A layup by P.J. Tucker put Houston up by three with about 2 minutes left and the Warriors missed shots on their next two possessions. That set up the 3-pointer by Harden with 49 seconds left that made it 124-118.
Kevin Durant made three free throws after that, but Harden added a layup to stretch the lead. Stephen Curry, who dislocated a finger in Game 2, missed a wide-open layup and Harden grabbed the rebound to secure Houston’s victory.
The Rockets withstood a 46-point performance from Durant and late surge by the Warriors to avoid falling into a 3-0 hole in the best-of-seven series.
Game 4 is Monday night in Houston.
Harden led the team despite recovering from injuries to both eyes he sustained in Game 2 when he was hit in the face by Draymond Green. He was feeling better on Saturday, but both eyes remained bloody red in spots.
“That’s James. That’s what he does,” Houston coach Mike D’Antoni said.
Eric Gordon added a playoff career-high 30 points with a playoff-best seven 3-pointers for Houston.
“We got outplayed. I thought Houston played really well. They deserved to win,” Warriors coach Steve Kerr said.
Curry finished with 17 points on 7-for-23 shooting after missing two layups in overtime. Green added 19 points, 11 rebounds and 10 assists for the Warriors.
Houston was clinging to a one-point lead when Iguodala’s 3-pointer put Golden State up 112-110 with 45 seconds remaining in the fourth. Harden made two free throws a few seconds after that to tie it and Durant missed a jump shot on the other end to give Houston a chance to win it in regulation.
Chris Paul was in the lane when Klay Thompson forced a jump ball. Harden got the ball but didn’t have time to get a shot off before the buzzer to send it to overtime.
A dunk by Durant cut Houston’s lead to one midway through the fourth quarter, but Austin Rivers hit a 3-pointer for Houston soon after that to leave the Rockets up 108-104.
The Warriors regained the lead with about 3 1/2 minutes to go after five straight points by Thompson made it 109-108. But Paul put Houston back on top with two free throws soon after that.
The Warriors used a 15-2 run at the end of the third quarter and the beginning of the fourth to take a 94-93 lead with about 10 minutes remaining. Durant was unstoppable in that stretch, scoring all 10 of Golden State’s points to start the fourth.
Tucker, Houston’s defensive stopper who had been on the bench with four fouls, re-entered the game to try and help slow Durant down.
Durant cooled off after that, but Iguodala and Thompson both made 3-pointers soon after that and the Warriors remained up by a point with about eight minutes left in the fourth.
The Rockets were up by nine at halftime, and Harden and Paul made 3-pointers on consecutive possessions in the third to push the lead to 72-61.
A 3-pointer by Curry cut Houston’s lead to seven later in the period before Houston used an 8-2 spurt to extend it to 83-70 with about 3 minutes left in the quarter.
Houston had a 12-point lead after a dunk by Clint Capela with less than a minute left in the third. Golden State scored the last five points of the quarter, with a 3 from Green, to get within 91-84 entering the fourth.
Houston got out to a much better start on Saturday night after being outscored 57-39 in the opening quarter of the first two games and the Rockets only trailed by one entering the second.
They used a 33-point second quarter, powered by 13 points from Gordon, to take a 58-49 lead into halftime.
Warriors: Durant passed Dwyane Wade for 10th place on the NBA’s career playoff scoring list. … Iguodala scored 16 points. … Golden State made 14 of 33 3-pointers.
Rockets: Gordon has made at least three 3-pointers in all eight playoff games this year. … Capela had with 13 points and 11 rebounds. … Paul had 14 points, eight rebounds and seven assists. … Tucker led the team with 12 rebounds and had three assists, two blocks and seven points.
After Monday’s game the series shifts to California where Game 5 is on Wednesday night.