The Patriots trailed 24-0 at halftime but came to life in the second half with Brady striking with 31 straight points to set up an exciting overtime finish.
When a New England punt struck Broncos player Tony Carter and was recovered by the Patriots at Denver's 13 yard-line, it allowed Stephen Gostkowski to kick the winning 31-yard field goal.
"It was nice to get a good bounce, and we needed it," Brady told reporters after the triumph. "We didn't give ourselves much of a chance in the first half."
It was a familiar outcome in the duel between the quarterbacks as Brady improved to 10-4 in matchups against Manning.
He threw for 344 yards and three scores to improve the Patriots to 8-3, one game behind the Broncos (9-2) and Kansas City Chiefs (9-2) in the AFC race.
Manning endured his worst game of the season as he completed just 19 of 36 passes for 150 yards, two scores and one interception.
Denver leaned on running back Knowshon Moreno, who exploded for 224 rushing yards and a TD, in building a lead.
New England lost two early fumbles that Denver converted into points, and Manning found Jacob Tamme for a 10-yard pass in the second quarter to establish the 24-0 advantage.
A determined Brady came out of the locker room and caught fire, however.
New England scored on its first five possessions of the second half to move ahead 31-24, with Brady throwing two touchdowns to Julian Edelman and one to Rob Gronkowski.
Those two players combined for 200 yards receiving.
Manning fought back and engineered a drive that saw him hit Demaryius Thomas on an 11-yard TD that tied the game with 3:06 left in regulation.
Both teams punted twice in overtime with New England catching the big break on their second punt.
"We just didn't do a good job protecting the ball in that second half," Manning said. "We just can't do that to our defense."
Denver receiver Wes Welker returned to New England where he spent his previous six seasons, and was booed while catching four passes for 31 yards.
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