Dallas didn't lead until Nowitzki hit two free throws 16 seconds into overtime, needing to rally from a 99-84 deficit in the final 5 minutes of regulation. The Mavericks never let the Thunder -- who were one win shy of tying an NBA record with eight OT wins in the regular season -- go ahead in the extra period.
Kevin Durant missed a 3-pointer on Oklahoma City's opening possession of overtime then didn't get another shot until he missed a 3 off the front rim in the final 10 seconds with the Thunder down by five.
Durant finished with 29 points and 15 rebounds, and Serge Ibaka had 18 points and 10 boards for Oklahoma City. Russell Westbrook added 19 points, eight rebounds and eight assists.
Game 5 is Wednesday night in Dallas.
Only two teams have come back from 3-1 deficits in NBA history without the benefit of home-court advantage in Game 7 -- Houston in the 1995 West semifinals and Boston in the 1968 East finals.
Durant had nine of the Thunder's 26 turnovers, including the one that led to Kidd's big shot.
Kidd stripped him as he went up for a shot with just over a minute left in overtime, then took a pass from Nowitzki, pump-faked to get Westbrook in the air and stepped up and drilled a 3-pointer to put Dallas up 108-105 with 40.3 seconds left.
Westbrook missed on a drive on Oklahoma City's next possession, and Jason Terry hit two free throws with 13 seconds left to give the Mavs a two-possession lead. Durant finally got another shot off, squatting with his head hanging down as Kidd walked up for two free throws to provide the final margin.
Terry finished with 20 points for Dallas and Kidd scored 17 to go with seven assists, five rebounds and four steals.
Fueled by the disappointment of losing Game 3 and squandering home-court advantage, the Thunder took control right away and never trailed in regulation.
With much of the sellout crowd remaining on its feet, the Thunder grabbed control with two separate bursts of seven straight points early in the fourth quarter. Durant had a two-handed slam and a 3-pointer off an offensive rebound in the second run, gesturing as though he were slapping on a pro wrestling championship belt after the 3 made it 99-84 with 5:06 remaining.
He hadn't won anything yet, though, especially with guard James Harden fouling out just after Dallas starting the closing run in regulation.
The Mavericks limited the Thunder to just one basket the rest of the way to wipe away the lead and tie it at 101 on Nowitzki's two free throws with 6.4 seconds left.
Nowitzki scored 12 points during the Mavs' 17-2 run and got fouled by Nick Collison, who grabbed the big German's hip as he lunged to tip the ball away, before hitting both foul shots to tie it.
Shawn Marion blocked Durant's 3-pointer at least 30 feet from the basket with 2 seconds left, and the Mavs couldn't convert a chance at the win when Kidd's inbounds lob with 0.7 seconds left hit the rim.
Oklahoma City came roaring out of the gates after trailing by as many as 17 points in the first quarter of Game 3. The Thunder hit their first nine shots and took an 18-8 lead after Durant caught a deflected inbounds pass and zoomed in for a right-handed jam.
Dallas managed to stay within striking distance, never falling more than 12 behind, and used repeated trips to the foul line to get within 59-54 at halftime. Nowitzki's driving layup cut the deficit to 61-59 with 8:53 left in the third quarter, and Kidd missed a jumper for the tie before Oklahoma City started pulling away again.
Durant provided a five-point possession during a 13-4 run that bumped the lead to 74-63 after Collison's two-handed dunk with 4:49 left in the third. He hit the first of two free throws, then added a 3-pointer after Collison got the rebound and kicked it out to him. Finally, Durant hit the free throw after Tyson Chandler was called for a technical foul for elbowing Kendrick Perkins while maneuvering for rebounding position on Durant's 3.
The Mavericks got within 79-77 in the final minute of the third quarter after a string of seven straight points that included a 3-pointer from Terry that made Thunder coach Scott Brooks turn away and put his head down on the scorer's table.
The NBA rescinded Chandler's first two technicals in this series, so his postseason count is currently at four -- three shy of what's needed for a one-game suspension. ... Dallas was the only visiting team to win twice in the regular season at the Oklahoma City Arena, where the Thunder were 30-11. The Mavs are 4-0 in the building in the regular season and playoffs. ... Brooks, facing repeated questions about his starting lineup, says he's sticking with it. "We're a young team," he said, "and if you give a young team instability, you're going to get very inconsistent results."