HOUSTON — Add the greatest comeback in Super Bowl history – and, considering the stakes, perhaps in NFL annals – to the New England Patriots’ astounding list of accomplishments.

On a night that was supposed to be about Deflategate vengeance or about reaffirming all-time greatness, the Patriots checked those boxes and far, far more. They used a rally for the ages, led by quarterback Tom Brady, to augment their Lombardi Trophy collection by scoring the game’s final 31 points to defeat the Atlanta Falcons, 34-28, in overtime here Sunday evening in a remarkable Super Bowl LI.

“They’re all sweet,” Brady said. “They’re all different. This has been just an incredible team, and I’m just happy to be part of it. . . . We overcame a lot of things, and it’s all worth it.”

The Patriots trailed 21-0 in the first half, 28-3 in the third quarter and 28-9 entering the fourth quarter. Yet they secured their fifth Super Bowl triumph in seven appearances in the big game with Bill Belichick as their coach and Brady as their quarterback.

Brady, in a season that began with a four-game suspension for his alleged role in the Patriots’ supposed scheme to use underinflated footballs in the first half of the AFC title game two years ago, moved past Joe Montana and Terry Bradshaw for the most Super Bowl victories by a quarterback. Belichick moved ahead of Chuck Noll for the most such wins by a head coach.

NFL Commissioner Roger Goodell was booed loudly when he handed the trophy to Patriots owner Robert Kraft on the field at NRG Stadium during the confetti-soaked aftermath. Goodell also shook Brady’s hand on the field.

“A lot has transpired during the last two years. . . . This is unequivocally the sweetest,” Kraft said to loud cheers.

No team ever had come from more than 10 points behind to win a Super Bowl. And no Super Bowl ever had gone into overtime.

This game produced both feats. The Patriots got the ball first in overtime and drove to tailback James White’s two-yard touchdown run 3:58 into the extra period.

“It’s just a lot of mental toughness by our team,” Brady said. “We’re going to remember this the rest of our life.”

Brady threw for 466 yards and two touchdowns on 43-for-62 passing and was named the game’s MVP. White had 14 catches for 110 yards and a touchdown and ran for two touchdowns.

Falcons quarterback Matt Ryan threw for 284 yards and two touchdowns.

“We made a few mistakes,” Ryan said, “and against a good football team like New England, those mistakes ended up costing us. . . . There’s nothing you can really say. It’s a tough loss, obviously.”

For most of the night, the Patriots mixed turnovers with spotty pass protection, no running game and dropped passes on offense. Tailback LeGarrette Blount lost a first-half fumble, and Brady threw an interception that was returned 82 yards for a touchdown by Atlanta cornerback Robert Alford. The Falcons got a touchdown run by tailback Devonta Freeman and touchdown passes by Ryan to tight end Austin Hooper and running back Tevin Coleman.

But Brady threw a third-quarter touchdown pass to White. The Patriots got closer with a field goal by kicker Stephen Gostkowski in the fourth quarter, then converted a fumble by Ryan on linebacker Dont’a Hightower’s sack into Brady’s six-yard touchdown pass to wide receiver Danny Amendola. After a successful two-point conversion, they were within eight points with just less than six minutes remaining.

They got the ball back and, aided by an otherworldly juggling catch by wide receiver Julian Edelman, got even with 57 seconds left on White’s one-yard touchdown run and another two-point conversion.

“We just kept plugging away, just kept fighting,” Belichick said.

Brady was sacked five times on the night. Three of the sacks were by Falcons defensive tackle Grady Jarrett. Atlanta ranked 25th in the league in total defense during the regular season. But the Falcons are young and fast on defense, and they dominated Brady and the New England offense for a half and a bit more.

“We went to the locker room at halftime,” White said. “We said the game isn’t over.”

Former President George H.W. Bush and wife Barbara arrive on the field for the coin flip. Charlie Riedel/Associated Press

After a scoreless first quarter, the Falcons started clicking on offense early in the second quarter with back-to-back completions from Ryan to wide receiver Julio Jones. Freeman provided a five-yard touchdown by bouncing outside to the left and leaping into the end zone.

The Falcons got the ball back and made it 14-0. Ryan found Hooper matched up with Patriots safety Patrick Chung on the left side and threw a dart of a pass for a 19-yard touchdown.

The Patriots, aided by three defensive holding penalties on the Falcons, moved into position to get back into the game. But on a third-and-six play from the Atlanta 23-yard line, Brady forced a short throw intended for Amendola into heavy traffic. Alford made the interception, veered toward the sideline to his right and had only Brady between him and the opposite end zone. He easily eluded the quarterback and could stroll at the end of his touchdown return.

The Patriots regrouped somewhat to get a 41-yard field goal by Gostkowski just before halftime. But it was more of the same in the third quarter. The Patriots got a stop on defense. But their drive ended with a drop by Edelman, and the Falcons responded with Ryan’s six-yard touchdown to Coleman.

“There was a lot of [stuff] that happened tonight,” Brady said.

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