The Patriots trailed 31-28 when Gostkowski recorded his first 53-yarder following a nine-play, 40-yard New England drive, forging a 31-31 tie.
He made his second long kick with 3:15 left, providing the margin of difference for the Patriots (9-3), who bested Denver by an identical 34-31 score last week -- a game in which they came back from a 24-point deficit.
Tom Brady completed 29-of-41 passes for 371 yards and two touchdowns.
Rob Gronkowski made six catches for 127 yards with a score and Julian Edelman added 101 receiving yards on nine catches.
Ben Tate scored three touchdowns en route to 102 rushing yards on 22 carries for the Texans (2-10).
Andre Johnson hauled in eight passes for 121 yards.