WASHINGTON — Triumphant over the signing of their far-reaching $1.9 trillion stimulus package, Democrats are now starting to angle for a major political payoff that would defy history: Picking up House and Senate seats in the 2022 midterm elections, even though the party in power usually loses in the midterms.

Democratic leaders are making one of the biggest electoral bets in years — that the stimulus will be so transformational for Americans across party lines and demographic groups that Democrats will be able to wield it as a political weapon next year in elections against Republicans, who voted en masse against the package.

Republicans need to gain only one seat in the Senate and just five in the House in 2022 to take back control, a likely result in a normal midterm election, but perhaps a trickier one if voters credit their rivals for a strong American rebound.

Yet as Democrats prepare to start selling voters on the package, they remain haunted by what happened in 2010, the last time they were in control of the White House and both chambers of Congress and pursued an ambitious agenda: They lost 63 House seats, and the majority, and were unable to fulfill President Barack Obama’s goals on issues ranging from gun control to immigration.

It has become an article of faith in the party that Mr. Obama’s presidency was diminished because his two signature accomplishments, the stimulus bill and the Affordable Care Act, were not expansive enough and their pitch to the public on the benefits of both measures was lacking. By this logic, Democrats began losing elections and the full control of the government, until now, because of their initial compromises with Republicans and insufficient salesmanship.

“We didn’t adequately explain what we had done,” President Biden told House Democrats this month about the 2009 Recovery Act. “Barack was so modest, he didn’t want to take, as he said, a ‘victory lap.’”

Now they are determined to exorcise those old ghosts by aggressively promoting a measure they believe meets the moment and has broader appeal than the $787 billion bill they trimmed and laced with tax cuts to win a handful of Republican votes in Mr. Obama’s first months in office.

Republicans say the Democratic bet is a foolhardy one, both because of how little of the spending is directly related to the coronavirus pandemic and because of fleeting voter attention spans. But Democrats say they intend to run on the bill — and press Republicans over their opposition to it.

“This is absolutely something I will campaign on next year,” said Senator Raphael Warnock of Georgia, who may be the most vulnerable incumbent Senate Democrat in the country on the ballot in 2022. Senator Gary Peters of Michigan, who heads the Democratic Senate campaign arm, said he would go on “offense” against Republicans who opposed the bill and sketched out their attack: “Every Republican said no in a time of need.”

Party lawmakers point out that the measure Mr. Biden signed on Thursday is more popular than the 2009 bill, according to polling; contains more tangible benefits, like the $1,400 direct payments and unemployment benefits; and comes at a time when the pandemic and former President Donald Trump’s continued appetite for big spending have blunted Republican attacks.

“People are going to feel it right away, to me that’s the biggest thing,” said Representative Conor Lamb, a Pennsylvania Democrat whose 2018 special election victory presaged the party’s revival. “Politics is confusing, it’s image-based, everyone calls everyone else a liar — but people are going to get the money in their bank accounts.”

And, Representative Sara Jacobs of California said, Democrats have “learned the lessons from 2009, we made sure we went back to our districts this weekend to tell people how much help they were going to get from this bill.”

Mr. Obama’s aides are quick to note that they did promote their stimulus and the health care law but ran into much more fervent, and unified, opposition on the right as the Tea Party blossomed and portrayed the measures as wasteful and ill-conceived.

At the end of last week, with the House’s first extended recess looming at month’s end, Speaker Nancy Pelosi pushed House Democrats to seize the moment.

Ms. Pelosi’s office sent an email to colleagues, forwarded to The Times, brimming with talking points the speaker hopes they’ll use in town halls and news conferences. “During the upcoming district work period, members are encouraged to give visibility to how the American Rescue Plan meets the needs of their communities: putting vaccines in arms, money in pockets, workers back on the job and children back in the classroom safely,” it said.

For their part, White House officials said they would deploy “the whole of government,” as one aide put it, to market the plan, send cabinet officers on the road and focus on different components of the bill each day to highlight its expanse.

Democrats’ hopes for avoiding the losses typical in a president’s first midterm election will depend largely on whether Americans feel life is back to normal next year — and whether they credit the party in power for thwarting the disease, despair and dysfunction that characterized the end of Mr. Trump’s term.

If voters are to believe the Democrats are delivering on an American rebound, of course, it’s essential the country is roaring back to prepandemic strength in a way it was not at the end of 2009, when unemployment reached 10 percent.

“You could be looking at an extraordinary growth spurt in the third and fourth quarters, and that takes you into the year when candidates make their way,” said Representative Richard E. Neal of Massachusetts, chairman of the Ways & Means Committee, where much of the bill was crafted.

The politics of the legislation, in other words, will be clear enough by this time next year. “If all the sudden you got high inflation and things are hitting the fan, Republicans are going to run on it,” said Representative Filemon Vela, a Texas Democrat. “If things are going well they’re going to run on something else.”

For now, Republicans are expressing little appetite to contest a measure that has the support of 70 percent of voters, according to a Pew survey released last week.

Part of their challenge stems from Mr. Trump’s aggressive advocacy for $2,000 direct payments in the previous stimulus package late last year, a drumbeat he’s kept up in his political afterlife as he argues Republicans lost the two Georgia Senate runoffs because they did not embrace the proposal.

It’s difficult for congressional Republicans to portray one of the main elements of the Democrats’ bill as socialism when the de facto leader of their party is an enthusiastic supporter of wealth redistribution. Moreover, right-wing media outlets have been more focused on culture war issues that are more animating to many conservatives than size-of-government questions.

Asked if they would run against the bill next year, the House minority leader, Kevin McCarthy, said, “There’s going to be a lot of things we run against.”

At the weekly news conference of House Republican leaders, Representative Liz Cheney of Wyoming spoke about the stimulus for 45 seconds before changing the subject to the rising number of migrants at the Southern border.

And by the end of the week, Mr. McCarthy announced he and a group of House Republicans would travel to the border on Monday in a bid to highlight the problem there — and change the subject.

After spending the campaign vowing to find common ground with Republicans and make Washington work again, Mr. Biden, in his first major act as president, prioritized speed and scale over bipartisanship.

He and his top aides believe in legislative momentum, that success begets success and that they’ll be able to push through another pricey bill — this one to build roads, bridges and broadband — because of their early win on Covid-19 relief.

“The fact that we could do it without Republicans forces them to the table,” said a senior White House official, who was not authorized to speak publicly about the nitty-gritty of lawmaking.

Yet to the G.O.P. lawmakers who have signaled a willingness to work with the new administration, Mr. Biden’s determination to push through the stimulus without G.O.P. votes will imperil the rest of his agenda.

“What I would be worried about if I were them is what does this do to jeopardize bipartisan cooperation on other things you want to do — you can’t do everything by reconciliation,” said Senator John Cornyn of Texas, alluding to the parliamentary procedure by which the Senate can approve legislation by a simple majority. “I’ve heard some of our members say that, ‘If you’re going to waste all this money on unrelated matters, I’m really not interested in spending a bunch more money on infrastructure.’”

To Senator Shelley Moore Capito of West Virginia, who was one of the Senate Republicans who went to the White House last month pitching a slimmed-down stimulus, it’s downright bizarre to hear Democrats claiming their 2010 difficulties stemmed from not going big.

“I would argue it was too big, it was unfocused, it was wasted money,” Ms. Capito said.

To Democrats, though, they are avoiding, not repeating, their past mistakes.

“The public didn’t know about the Affordable Care Act and the administration was not exactly advertising,” Ms. Pelosi told reporters last week.

Senator Chuck Schumer, the majority leader, was just as blunt, singling out the Maine moderate who was wooed by Mr. Obama to ensure bipartisan support for the 2009 Recovery Act but whose appeals for a far-smaller compromise bill were ignored last month.

“We made a big mistake in 2009 and ’10, Susan Collins was part of that mistake,” Mr. Schumer said on CNN. “We cut back on the stimulus dramatically and we stayed in recession for five years.”

And, he could have noted, his party would not have full control of both ends of Pennsylvania Avenue for another decade.