Eric Gordon helped the Rockets get to that point with the best playoff performance of his career.
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Harden scored 41 points, and Houston outlasted the Warriors 126-121 on Saturday night to cut the deficit in the Western Conference semifinals to 2-1.
WARRIORS-ROCKETS: Western Conference playoff schedule
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 the victory.
Gordon added a playoff career-high 30 points with a playoff-best seven 3-pointers.
"It's all about bringing something to the table, offensively, defensively every single night to give yourself a chance to win," he said.
Durant said Gordon was "very important" to Houston's win.
"He kept them afloat for a while," he said.
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 said they didn't bother him on Saturday, but both eyes remained bloody red in spots.
"That's James. That's what he does," Houston coach Mike D'Antoni said.
Curry finished with 17 points on 7-for-23 shooting after missing two layups in overtime. He wouldn't say if his injury was limiting him on Saturday night.
"I've just got to make those," he said. "If I'm out there playing, I've got to produce and it just didn't happen tonight."
Green added 19 points, 11 rebounds and 10 assists.
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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.
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. Tucker was really unhappy while he sat on the bench and watched Durant singlehandedly give Golden State the lead.
Asked if he said something to D'Antoni about it to get back in the game he said: "Oh, I said something to all of them."
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 8 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.
TIP-INS
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.
TUCKER'S VALUE
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D'Antoni and Kerr as well as players from both teams raved about the job Tucker did.
"We say it all the time, Tuck doesn't do any of the glamorous stuff," Paul said. "All the stuff he does doesn't show up on the stat sheet ... but it's always nice for other people to see it and appreciate it."
Kerr said Tucker's offensive rebounding down the stretch was a key.
"The plays P.J. Tucker made in overtime and the fourth quarter, those were big plays," Kerr said. "Those are indicators of who played harder."
Said D'Antoni: "Tuck would not let us lose."
HARDEN'S EYES
When Harden was asked if his eyes bothered him considering how they looked on Saturday night he answered with a straight face.
"What do you mean, how they look?" he asked.
Paul jumped in quickly with the answer.
"They look crazy," he said laughing before Harden snickered, too.
UP NEXT
After Monday's game the series shifts to California where Game 5 is on Wednesday night.
See also:Warriors-Rockets II: Everything to know for the must-see sequel
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