Ex-Patriot Chandler Jones sacked Garoppolo on the first play of New England's next possession, but the young quarterback still set up the winning kick by Gostkowski, who also booted a 53-yarder.
New England, a five-point underdog, took the lead on its first possession and scored again to start the second half.
Up 10-7 at the break, the Patriots took the second-half kickoff and went 70 yards in nine plays. Rookie cornerback Brandon Williams was beaten badly for the second time on Garoppolo's 28-yard pass to the Arizona 8. LaGarrette Blount bulled it in from there and the Patriots led 17-7 with 10 minutes left in the first quarter.
The first two Arizona touchdowns followed New England turnovers.