New England benefited from two critical replay reviews and made their ninth Super Bowl with Brady at quarterback and Bill Belichick as coach.
"This is crazy," said Brady, who was 30 of 46 for 348 yards. "What a game."
It's the first time both conference title games went to OT. The last time both visitors won conference championship matches was 2012; New England was 3-5 on the road this season.
"We knew what our record was but we didn't let that dictate us," said defensive end Trey Flowers, who led a staunch charge on Patrick Mahomes. "Whatever happened in the regular season happened. We came out and did it when it counted."
Several times, the Patriots appeared to have it won, only to see Kansas City (13-5) come back in spectacular fashion.
Brady, at 41 already the oldest quarterback to have played in a Super Bowl, drove New England 65 yards in 1m 24s to Burkhead's go-ahead 4-yard touchdown with 39s left in regulation. That was enough, though, for his far younger counterpart, the 23-year-old All-Pro Mahomes, to take the Chiefs 48 yards to Harrison Butker's 39-yard field goal with 8s left to force overtime.
It was a sizzling offensive showing in the fourth quarter after defence had been in charge most of the way. Indeed, the Chiefs were blanked in the opening half for the first time all season.
And they never saw the ball in overtime, which along with the two replay decisions might call into play NFL rules and officiating.
"I thought if we got the chance," Mahomes said, "we'd score."
Mahomes finished 16 of 31 for 295 yards and three touchdowns.
New England became the third franchise to reach three Super Bowls in a row.
- AP