But Gostkowski, who had made his other four field-goal attempts, sent a 42-yard try wide to the left.
The Cardinals (2-0) won behind Kolb, filling in for John Skelton, who missed the game with a sprained right ankle. The Patriots (1-1) lost tight end Aaron Hernandez to an ankle injury in the first quarter.
Kolb overcame a shaky start, then threw a 2-yard touchdown pass to Andre Roberts 6 minutes into the third quarter after a blocked punt and ran for a 5-yard score on the third play of the fourth as the Cardinals took a 20-9 lead.
Trailing 20-12 after Gostkowski's 53-yard field goal, Brady led his best drive of the game. He completed eight of 10 passes for 82 yards, capped by a 5-yard touchdown throw to Gronkowski, making it 20-18 with 2:06 left. But his pass to Gronkowski for a 2-point conversion was broken up by Kerry Rhodes.
The loss was the first in 11 home openers since the Patriots moved into Gillette Stadium in 2002. The Cardinals ended a five-game losing streak against the Patriots, beating them for the first time since Sept. 29, 1991.
The Patriots made numerous uncharacteristic mistakes. Patrick Peterson intercepted Brady on the Patriots' first offensive play, the Cardinals sacked him four times and they blocked a punt by Zoltan Mesko, the first time that happened to him in 122 punts as a pro.
That block gave Arizona the ball at the Patriots 2-yard line. After Williams ran for no gain and Kolb threw an incompletion, the quarterback hit Roberts on a quick slant for the touchdown.
The teams were tied at 6 after a lackluster half in which Brady passed for just 75 yards and Kolb for 59.
Arizona wasted an excellent opportunity on the game's first series when it settled for Jay Feely's 38-yard field goal after marching 60 yards on 12 plays. He added a 47-yarder after the Cardinals failed to get a first down on the series after Peterson's interception.
Then the Patriots stalled after moving from their 25-yard line to the Cardinals 28, coming away with only a 46-yard field goal by Gostkowski. He tied the game with a 29-yarder with 9:08 left in the first half.
Brady completed 28 of 46 passes for 316 yards and extended his streak to 34 games with at least one touchdown pass, fourth-best in NFL history. Kolb was just 15 for 27 for 140 yards but was sacked only once, threw no interceptions and made several good passes while rolling out.
Arizona 20, New England 18
Arizona 6 0 7 7 20
New England 3 3 3 9 18
Ari • FG Feely 38, 8:42.
Ari • FG Feely 47, 7:42.
NE • FG Gostkowski 46, 2:22.
NE • FG Gostkowski 34, 9:08.
NE • FG Gostkowski 51, 12:53.
Ari • Roberts 2 pass from Kolb (Feely kick), 8:52.
Ari • Kolb 5 run (Feely kick), 14:00.
NE • FG Gostkowski 53, 6:45.
NE • Gronkowski 5 pass from Brady (pass failed), 2:06.
Attendance • 68,756.
First downs 16 25
Total Net Yards 245 387
Rushes-yards 33-105 28-90
Passing 140 297
Punt Returns 1-3 2-25
Kickoff Returns 2-50 2-46
Interceptions Ret. 1-0 0-0
Comp-Att-Int 15-27-0 28-46-1
Sacked-Yards Lost 1-0 4-19
Punts 6-46.7 6-28.7
Fumbles-Lost 2-2 0-0
Penalties-yards 5-39 8-60
Time of Possession 28:19 31:41
Rushing • Arizona, Wells 14-44, Kolb 6-20, Peterson 1-17, R.Williams 10-13, Stephens-Howling 2-11. New England, Ridley 18-71, Woodhead 8-18, Hilliard 1-2, Brady 1-(minus 1).
Passing • Arizona, Kolb 15-27-0-140. New England, Brady 28-46-1-316.
Receiving • Arizona, Heap 5-62, Doucet 3-21, R.Williams 2-10, Wells 1-24, Stephens-Howling 1-10, Housler 1-7, Fitzgerald 1-4, Roberts 1-2. New England, Lloyd 8-60, Gronkowski 6-75, Welker 5-95, Edelman 5-50, Ridley 3-24, Woodhead 1-12.
Missed Field Goals • New England, Gostkowski 42 (WL).