The Patriots trailed by 13 points in the second half. They took their first lead with 15 seconds to play on a 21-yard Nick Folk field goal.
New England left Houston with a 25-22 win.
Folk kicked field goals of 52, 52 and 32 yards in addition to his game-winner.
The Patriots trailed 22-9 after Lonnie Johnson picked off Mac Jones on the second play of the second half and Davis Mills hit Chris Conley for a 37-yard touchdown strike off a flea flicker.
But it was all New England after that.
Jones finished 23-of-30 for 231 yards with one touchdown and an interception. Henry caught six passes for 75 yards, and his 13-yard touchdown catch with 9:31 remaining tied the game.
Damien Harris twice left the game with a chest injury and had a costly fumble into the end zone in the first half, a play that was ruled a touchdown on the field. Replay reversed it.
The Texans couldn’t get out of their own way. Cameron Johnston had a punt for 0 yards after hitting the back of a teammate’s helmet with the kick. Ka’imi Fairbairn missed two extra points in the first half and a a 56-yard field goal in the second half that set up the Patriots’ game-tying touchdown.
Davis Mills went 21-of-29 for 311 yards and three touchdowns. Chris Moore caught five passes for 109 yards and a touchdown.