HOUSTON — They played Super Bowls for 50 years before this one, and it may be another half-century before we see something like this again.

The New England Patriots, trailing 28-3 in the third quarter, with quarterback Tom Brady battered around by the Atlanta Falcons’ defense and also making some uncharacteristic bad throws, scored two touchdowns and a pair of two-point conversions in the final six minutes of the fourth quarter and another TD on the first possession of overtime to win the Super Bowl 34-28 Sunday night.

It was first overtime game in Super Bowl history, the Patriots’ fifth championship in seven Super Bowls for the tandem of Brady and coach Bill Belichick, and by far the largest comeback in Super Bowl history. The previous record rally was only 10 points.

Brady, forced to keep passing to fuel the comeback, finished the game with 466 yards passing, a Super Bowl record, to break a tie of four Super Bowl titles with Terry Bradshaw and Joe Montana, his boyhood idol. Brady completed 43 of 62 attempts with the two touchdowns and one interception, which was returned for a second-quarter TD.

Belichick became the first coach to win five Super Bowl titles, breaking a tie with Pittsburgh’s Chuck Noll.

“Two years ago, we won our fourth Super Bowl down in Arizona, and I told our fans that was the sweetest one of all,” Patriots owner Robert Kraft said. Then, in a not-so-subtle reference to the Deflategate controversy that caused NFL commissioner Roger Goodell to suspend Brady for the first four games of the season, Kraft added, “but a lot has transpired in the last two years.

“This is unequivocally the sweetest.”

How could it not be?

For much of the night, it looked as if the game would be over early. The Falcons, whose defense allowed more points than all but five NFL teams during the regular season, stymied Brady and the Patriots for three quarters. He was sacked five times.

Atlanta led 21-3 at halftime, and tacked on another touchdown in the third quarter for the 28-3 lead.

“At halftime, we came in and Tom was confident, calm, that goes for our whole team,” Patriots offensive coordinator Josh McDaniels said. “This group would never quit.”

And, as usual, the group was superbly prepared to go the distance.

“We worked on two-point plays all week,” Belichick said. “Josh and I, we thought we might need a couple of them.”

Two of them, as it turned out.

New England controlled the ball but couldn’t score for a long time, one drive ending with a fumble and another with a Brady interception that Robert Alford returned 82 yards for an Atlanta touchdown. Then came the fourth quarter. In just the fourth quarter and overtime, Brady completed 21 of 27 passes for 246 yards. He was selected as the Super Bowl MVP for a record fourth time.

“That was exactly the way we didn’t plan it,” Brady said. “Everything we didn’t want to do. But that was a great football game. … We’re all going to remember this for the rest of our life.”

The Patriots scored touchdowns on their last three possessions, not including a three-second possession at the end of the fourth quarter. Once they won the coin toss and took the ball in overtime, the result seemed pre-ordained, and it came on a 2-yard run by James White — his third TD of the game — after 3:58 of overtime, capping a 75-yard, eight-play drive.

Brady began the winning drive by completing five consecutive passes for 50 yards. A 10-yard run by White and a pass-interference penalty moved the ball to the 2-yard line before White’s winning run.

White caught a Super Bowl-record 14 passes (for 110 yards), and he added 29 yards on six rushes.

As terrific as the Patriots’ comeback was, Atlanta contributed to its own demise, and Falcons coach Dan Quinn admitted there was “no place to put that one mentally for us.”

First, with the Falcons ahead 28-12 and just under 10 minutes remaining in the fourth quarter, they refused to try to play clock ball even though they were averaging more than 5 yards a rush.

The strategy backfired when quarterback Matt Ryan, dropping deep to pass on third-and-1 in his own territory, was sacked by Dont’a Hightower and fumbled, and the loose ball was recovered by Alan Branch at the Falcons’ 25-yard line with 8:24 remaining.

From here, it took New England just five plays and 2:28 to score, the touchdown coming on a 6-yard pass from Brady to Danny Amendola. The PAT came on a run up the middle by White, who took a direct snap in the shotgun on a trick play.

That made it 28-20 with 5:56 remaining, enough time that the Patriots did not have to try an onside kick.

Here is where Atlanta made its second major blunder, coming after the Falcons got a 39-yard pass-and-run from Ryan to Devonta Freeman on the first play after the kickoff.

With 4:40 remaining, Atlanta had a first down at the New England 22-yard line, in position for a field goal that would have made it a 10-point difference and perhaps clinched the game. Instead, the Falcons had Ryan dropping back deep again, and this time, a Trey Flowers sack pushed them back to the 35-yard line.

A holding penalty ended any chance of even a long field goal, and the Falcons had to punt the ball back to New England with 3:30 remaining. Even though the Patriots started at their own 9-yard line after the punt, there was an inescapable feeling in the stadium that more magic was about to come from Brady, all seven of whose Super Bowl games were decided in the closing moments.

The 91-yard drive required 10 plays, the key one a remarkable circus catch by Julian Edelman, who was on the ground to grab a pass that was tipped by Alford. Edelman cradled the ball against a defender’s leg and gained 23 yards.

“That was one of the greatest catches I’ve ever seen,” Brady said of the Edelman grab. “I don’t know how he caught it. I don’t think he does.”

A 20-yarder to Amendola followed. After all that, it was almost anticlimactic when White scored on a 1-yard run with 57 seconds remaining. The two-point PAT, a pass from Brady to Amendola, tied the game.

Edelman’s catch was reminiscent of a stunning catch by David Tyree of the New York Giants late in the fourth quarter of Super Bowl XLII, a play that handed Brady one of his two Super Bowl defeats.

“The most important thing tonight is a team win,” Belichick said. “Chuck Noll is a tremendous coach, tremendous legacy. I always admired him. … it’s an honor to be mentioned in the same sentence as Chuck Noll.”

NOTES: The Patriots failed to score a first-quarter point in any of their seven Super Bowls with the combination of coach Bill Belichick and QB Tom Brady. … Pro Bowl C Alex Mack was active and in the starting lineup for the Falcons. Mack did not practice this week because of a left fibula injury, reportedly a chip fracture, which could have limited a player considered vital to Atlanta’s zone-running scheme. … WR Michael Floyd was inactive for the Patriots, who listed the same players who did not play in the AFC Championship Game on their pregame inactive list: third QB Jacoby Brissett, DB Cyrus Jones, OL LaAdrian Waddle, RB D.J. Foster and DB Justin Coleman and S Jordan Richards … WR Nick Williams. RB Terron Ward, CB Blidi Wreh-Wilson, LB Josh Keyes, S Dashon Goldson, OL Wes Schwetzer and TE D.J. Tialavea were inactive for the Falcons.