The 55-year-old shared his triumph with partner Oti Mabuse, the first Strictly dancer to win for two years in a row.
Bailey beat EastEnders' Maisie Smith and singer HRVY at the end of Saturday's grand final.
"It feel surreal, it feels extraordinary, it feels wonderful," Bailey said as he was named the winner.
"I never thought we'd get this far, never thought we'd get to the final.
"But I have had the most extraordinary teacher and the most extraordinary dancer," he added, paying tribute to Mabuse. "Someone who believed in me right from the beginning, and she found something in me and turned me into this, into a dancer."
In response Mabuse told him: "I think you are amazing, remarkable. You just put your heart and soul into everything. Thank you for being a friend, a father figure to me, a brother, and for this [the glitterball trophy]!"
Actor Joe McFadden had been Strictly's previous oldest winner, having won in 2017 at the age of 42.
Mabuse, who has danced on Strictly since 2015, also won last year's series with Emmerdale actor Kelvin Fletcher.
Aliona Vilani is the only other pro dancer to have triumphed twice, having won with Harry Judd and Jay McGuinness in 2011 and 2015 respectively.
Made in Chelsea's Jamie Laing also made it to this year's final, having survived an unprecedented four dance-offs throughout the series.
Bailey is known for his appearances on QI, Never Mind the Buzzcocks and the Channel 4 sitcom Black Books.
Analysis by Steven McIntosh, entertainment reporter
In an interview last week, Craig Revel Horwood said he "really thought Bill Bailey would be the Ann Widdecombe of this series".
And that perfectly sums up the attitude many had towards Bill at the beginning of Strictly 2020. At his age, particularly being a comedian, he would surely fall into the novelty category; hired for entertainment value rather than serious dancing.
But Bill gradually improved as the weeks went on, with his routine to Rapper's Delight by the Sugarhill Gang (who later praised his performance) proving a turning point. Viewers realised he was focused and really putting in the hours to learn complex routines.
"It makes me smile to have confounded people's expectations," Bailey recently wrote in The Telegraph. "I always intended to give it my all, perhaps to offset my pantomime horse role - but what I didn't expect was to be able to dance well, certainly not with a degree of confidence."
Bill Bailey was not the best dancer in this year's Strictly. Until tonight's final, he hadn't topped the leaderboard, often trailing behind the younger, more agile, contestants like Maisie and HRVY.
But that didn't matter. Being the best dancer is actually not what Strictly is about. Much more important is the journey a celebrity goes on over the series; their effort, their commitment, their improvement. Oti's continuing popularity certainly didn't hurt, but ultimately the British public loves an underdog.
Bailey became a firm fan favourite during his time on the show, particularly after his and Oti's dance to Rapper's Delight by The Sugarhill Gang, which went viral earlier in the series.
Their Couple's Choice routine was one of three the pair performed on Saturday night.
The pair also reprised their week two Quickstep to Bobby Darin's Talk to the Animals, as well as a new Showdance to Queen's The Show Must Go On.
Speaking ahead of the final, Bailey said it was "wonderful" if he had come to be seen as a role model for mature would-be hoofers.
"Blokes sometimes feel a bit self-conscious, particularly blokes of my age," the hirsute funnyman told journalists. "They feel like they're going to be called the dad dancer.
"I think if me showing I can get out there and look a little bit more than somebody shuffling about, then why not?"
This year's series was shorter than usual due to the coronavirus pandemic, though that did not prevent the virus having an impact.
HRVY tested positive for coronavirus 10 days before the launch show was filmed, while boxer Nicola Adams was forced to withdraw when her partner Katya Jones tested positive.
The couple had made Strictly history by becoming the first same-sex duo to compete on the programme. They returned for a special performance in Saturday's final.
There was also a musical performance from Robbie Williams and an appearance from the Duchess of Cornwall, who paid tribute to the cast and crew of Strictly for "lifting the whole country's spirits".
"I'd like to, on the behalf of everybody who watches Strictly, to say an enormous thank you to everybody," she said. "Everybody who has been involved in this production, in this particularity difficult year, you have given everybody so much pleasure and you've uplifted the nation."
Concerns over Transatlantic travel meant Bruno Tonioli could only appear virtually this year, while judge Motsi Mabuse - Oti's older sister - had to take two weeks off in order to self-isolate.
That was good news for Anton Du Beke who, having been eliminated in week two along with his partner Jacqui Smith, got to sit on the judging panel while Motsi was away.
The BBC received more than 150 complaints from viewers after three of the other professional dancers appeared in drag during a Priscilla, Queen of the Desert routine.
Claudia Winkleman, meanwhile, was forced to make an on-air apology after The Wanted's Max George was heard uttering a profanity after one of his dance routines.
Claudia and Tess Daly will be back on Christmas Day to present a Strictly special featuring 25 of the BBC One show's most memorable routines to date.
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