The potential winning kick was set up when linebacker Brandon Spikes forced Cardinals running back Ryan Williams to fumble with 61 seconds left and Vince Wilfork recovered at the Arizona 30-yard line.
After a would-be winning touchdown run by Danny Woodhead was called back due to a holding penalty on Rob Gronkowski, the Pats got to the ball to the 24.
But Gostkowski -- who had been 4-for-4 in the contest -- sent his attempt wide left and Arizona (2-0) hung on for the victory.
Kevin Kolb went 15-of-27 through the air for 140 yards and interception.
Five of his completions went to Todd Heap, who racked up 62 yards.
Tom Brady amassed 316 yards, a TD and an interception on 28-of-46 efficiency for the Patriots (1-1).