ATLANTA — Georgia’s secretary of state, Brad Raffensperger, on Tuesday certified the runoff election victories of Senators-elect Jon Ossoff and the Rev. Raphael Warnock, setting in motion the formal legal process that will seat the two Democrats and give their party control of the U.S. Senate for the first time since 2015.

The swearing-in of Mr. Ossoff, Mr. Warnock and Alex Padilla, who will fill the California Senate seat left vacant by Vice President-elect Kamala Harris, will create a 50-50 tie in the Senate, giving Democrats de facto control of the chamber because the tiebreaking vote will be held by Ms. Harris. She will be sworn in as vice president on Wednesday, and the three new Democratic senators are expected to be sworn in on Wednesday afternoon.

Gov. Brian Kemp of Georgia, who like Mr. Raffensperger is a Republican, also signed off on the certification of the races. Gabriel Sterling, a top official in Mr. Raffensperger’s office, noted on Twitter last week that a representative of Georgia state government must then go to Washington to hand the certification documents over to the secretary of the Senate.

Despite a flurry of recent drama and unfounded allegations of voter fraud in Georgia, there was little doubt that Mr. Raffensperger would eventually certify the results of the Jan. 5 contests in which Mr. Ossoff defeated David Perdue, a one-term Republican senator, and Mr. Warnock beat Kelly Loeffler, a Republican who was appointed to the Senate seat by Mr. Kemp in December 2019.

The margins in both races were outside the half-percentage point threshold that allows the trailing candidate to demand a statewide recount under Georgia law. With about 4.4 million ballots cast, Mr. Ossoff won his race by about 55,000 votes, giving him a 1.22 percent lead, and Mr. Warnock won by about 93,000 votes, giving him a 2.08 percent lead, according to the secretary of state’s website.

Those results stood in contrast to those of the Nov. 3 presidential election, in which Mr. Biden defeated Mr. Trump by a narrower margin that was well within the threshold, allowing Mr. Trump to demand a recount.

The recount in the presidential race showed that Mr. Trump had indeed lost by about 12,000 votes. But that did not stop the president and his allies from continuing to vigorously press the unfounded allegation that he was the victim of a rigged election.

That false narrative, which Mr. Trump pursued in failed court cases and in campaign appearances, quite likely ended up helping the two Democratic Senate candidates by depressing turnout in Georgia among those supporters of the president who saw no reason to vote in an electoral system that he was constantly maligning as untrustworthy.

The two Senate races presented a rare and remarkable drama in American politics, given Mr. Trump’s recalcitrance, Mr. Biden’s triumph and the effect that control of the Senate would most likely have on Mr. Biden’s initial policy agenda. Outside money poured into Georgia, making for the most expensive Senate races in U.S. history. Mr. Trump flew to the state and held big, well-attended rallies for Mr. Perdue and Ms. Loeffler. But his message of support was often overtaken by his compulsion to air grievances about his own election.

The two Democrats vowed to strengthen the Affordable Care Act, support police reform and overhaul the national response to the coronavirus pandemic. The two Republicans darkly warned that Democratic victories would hasten a dangerous national slide into radical socialism.