Mr Hancock was speaking to ITV’s Good Morning Britain when he apparently became overcome with emotion over the news, which he said made him “proud to be British”.
The 81-year-old namesake of the English playwright received his Covid-19 jab at University Hospital Coventry in Warwickshire, near the bard’s birthplace in Stratford-Upon-Avon.
The health secretary’s reaction received a mixed response on social media, with some users praising the minister and others suggesting he was faking tears for the camera.
“It’s just been such a tough year for so many people and there’s William Shakespeare putting it so simply for everybody that we can get on with our lives,” Mr Hancock told ITV’s Piers Morgan and Susanna Reid.
“There’s still a few months to go. I’ve still got this worry that we can’t blow it now, Piers.
“We’ve still got to get the vaccine to millions of people so we’ve got to keep sticking by the rules but there’s so much work that’s gone into this and it makes you so proud to be British.”
Ministers have previously been criticised for trying to present the Pfizer jab as a primarily British success story as it was developed in Germany, manufactured in Belgium and the US, and trialled in six countries around the world.
The government has also been keen to stress in recent days that the UK’s mass vaccination campaign will take months to complete and social distancing restrictions will likely remain in place until at least spring.
Boris Johnson said on Tuesday that the Pfizer vaccine and others which are seeking approval from the UK’s medicines regulator would make a “huge difference” in the pandemic.
However, he noted it would take time for the majority of the population to receive a vaccine.
“It will gradually make a huge, huge difference, but I stress ‘gradually’ because we are not there yet, we have not defeated this virus yet,” Mr Johnson said.
“We can’t afford to relax now and so my message would be it’s amazing to see the vaccine coming out.”
Mr Hancock told the BBC that he expected “several million” people to be vaccinated by Christmas but warned it was still unknown whether the Covid jab could prevent someone passing on the disease.
“We know that this vaccine protects the person who’s been vaccinated. What we don’t yet know is the degree to which the vaccine reduces the chance of [someone] passing it on … asymptomatically,” he told BBC Breakfast.
“Now, we think that that transmission risk, as it’s called, is much reduced by being vaccinated. But we need to watch that.”
He added that the government would release figures on the number of people who have received a jab over the coming weeks as officials monitor hospital admissions data to assess the vaccine’s effect on the pandemic.
Additional reporting by PA