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The Patriots trailed by 25 points in the third quarter but recovered to level at 28-28 and force the extra period - the first in Super Bowl history.
Thanks to everyone back in Boston... we're bringing this sucker home!
Quarterback Tom Brady led the recovery, finishing with a record 466 yards en route to being named the Super Bowl's Most Valuable Player for a fourth time.
The turnaround was completed when James White scored on a two-yard run - taking his personal haul for the game to 20 points.
The previous biggest deficit overcome by the eventual Super Bowl champions was 10 points, a record emphatically shattered by the Patriots on an incredible night at Houston's NRG Stadium, which also featured a spectacular half-time show by Lady Gaga.
The singer opened her set with Woody Guthrie's civil rights anthem This Land Is Your Land, "a gentle but pointed rebuke to the Trump administration", BBC Music reporter Mark Savage says.
Notable statistics from the game included:
Brady, 39, admitted afterwards the outcome could have been very different had any part of the Patriots team not done its job.
A key moment came with the Patriots trailing 28-20 with 2:28 remaining in the final quarter when Julian Edelman made a miraculous catch for a first down, somehow grabbing the ball under pressure from three opponents after it was tipped into the air by Falcons cornerback Robert Alford.
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"I couldn't believe the Edelman catch, it was one of the greatest catches. I don't think he knows how he caught it. We've been on the end of a few of those, it was spectacular," Brady said.
"It's going to be a great celebration tonight. Thanks to everyone back in Boston, we love you, we're bringing this sucker home!"
He added: "That was exactly the way we didn't plan it. It was a hell of a football game.
"This is an incredible team and I'm just happy to be a part of it. We overcame a lot of different things and it's all worth it."
"To be 28-3 down, it was a lot of mental toughness from our team and we're all going to remember this for the rest of our life."
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Much of the talk before the game centred on whether Brady could become the first quarterback to win five Super Bowls, but such thoughts were swiftly pushed to one side once the game began as the veteran struggled to find a rhythm.
The first quarter whipped by with hardly any stoppages and no points scored, both defences on top, but come the second quarter the momentum shifted emphatically in favour of Atlanta, who boasted the best regular-season offense and, in quarterback Matt Ryan, the NFL's MVP.
Atlanta went ahead when Devonta Freeman capped an impressive half by rushing for a touchdown, while Ryan connected with Austin Hooper for the second a short time later.
For the last year and a half I've talked about how Tom Brady is not the greatest of all time. I take it all back
Brady, meanwhile, was labouring, struggling to connect with his receivers and cutting an increasingly frustrated figure as half-time loomed.
When Alford intercepted Brady for an 82-yard touchdown, the writing appeared to be on the wall - with a 21-point lead already double the highest deficit ever overcome in a Super Bowl - although a late field goal at least gave the Pats a sliver of hope at the interval.
As Lady Gaga descended into the arena, Falcons fans were no doubt daring to dream that the franchise could break its Super Bowl duck at the second time of asking, having been beaten by the Denver Broncos on their only previous appearance in 1999.
But if they were thinking along those lines, they reckoned without Brady.
Certainly the omens looked good for Atlanta at the start of the second half with Tevin Coleman's score taking their advantage to 28-3, but that was the cue for New England's fightback to begin.
White scored what seemed like a consolation touchdown late in the third quarter, a feeling only heightened by Stephen Gostkowski's failed extra-point attempt, but still the Pats kept coming.
A Gostkowski field goal was backed up by Brady's touchdown pass to Danny Amendola, with the successful two-point conversion from a White rush closing the gap to eight points.
With Atlanta rattled, Brady marched the Patriots 90 yards upfield via Edelman's stunning catch to present White with a one-yard rushing touchdown, which was followed by a vital two-point conversion catch by Amendola that took the game to overtime.
Patriots quarterback Tom Brady: "There were a lot of plays, probably about 30 of them, and if any one was different the outcome would have been different. It was unbelievable. I'm so proud of these guys.
"James White is everything you want in a team-mate. Dependable, reliable, durable. He brings it every day. We kept going to him and that speaks for itself."
Patriots wide receiver Danny Amendola: "[Brady] was the same as he always is: cool, calm and collected. He's the leader, the general, the best ever and that is the end of the story."
Patriots running back James White: "We knew we had a shot the whole game. It was an amazing comeback by our team. It's surreal right now. You couldn't write this script."
Patriots coach Bill Belichick: "We have great players, they competed the whole game. They were 28-3 down but they never looked back. They just keep competing for 60 minutes, or longer."
Patriots owner Robert Kraft: "I told our fans two years ago that was the sweetest win of all, but a lot has transpired in the last two years. That doesn't need any explanation.
"This is unequivocally the sweetest. I am proud to say for the fifth time the Patriots are world champions."
Falcons quarterback Matt Ryan: "There's nothing you can really say. That's a tough loss. Obviously very disappointed, very close to getting done what we wanted to get done, but it's hard to find words tonight."
Osi Umenyiora, BBC NFL analyst: "There is no solace for Atlanta. They were ahead by 25 points, a game they had to win. I can't imagine how they are feeling. They made so many young mistakes, you can't make them against the New England Patriots.
"This is the worst loss we have ever seen in the history of the Super Bowl.
"For the last year and a half I've talked about how Tom Brady is not the greatest of all time. I take it all back. The improbability of what this man just did, I can't believe what I have seen with my own two eyes.
"It is unbelievable. I take back every negative thing I ever said about this man, he shut me up today. He truly is the greatest."
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The New England Patriots produced the greatest comeback in Super Bowl history to beat the Atlanta Falcons 34-28 in overtime and claim a fifth title in the most dramatic of circumstances.