Gasol pumped both fists after the basket, and the Lakers moved on when Russell Westbrook missed a desperation 3-pointer at the final buzzer.
Kevin Durant led the Thunder with 26 points on a dreadful 5-for-23 shooting night. Westbrook had 21 points and nine assists.
Bryant scored 32 points and Gasol had nine points and 18 rebounds, scoring his only basket of the second half at the end.
After missing 17 of his first 20 shots, Durant broke through with a 3-pointer from the right wing that ignited a 10-0 comeback by the Thunder. Westbrook followed with a jumper, and his three-point play off a putback of his own miss put Oklahoma City ahead 92-91 with 3:04 to play.
Durant added a driving layup to push the lead to three but Oklahoma City couldn't score again in the final 2½ minutes.
Bryant hit an 18-foot jumper from the left side to get Los Angeles within one, and the Lakers delivered the knockout punch on their fifth chance.
Los Angeles advances to face the winner of the Denver-Utah series.
The extended series took its toll on a Los Angeles squad that already was nursing more than its fair share of injuries. Andrew Bynum played the second half with a brace on his right knee after he hyperextended it earlier in the game. Lamar Odom had a brace on his right knee, too. Ron Artest sported a modified T-shirt with padding on his left shoulder that's gotten a workout against Durant, the youngest player to win the NBA scoring title.
Bryant has a sore knee, ankle and index finger. Bynum just got back from a strained Achilles' tendon that cost him the final 13 games of the regular season.
The Lakers may be battered but not beaten.
Bryant almost single-handedly kept Los Angeles out front with 16 points in a dazzling third quarter - until it came to an early end for him when he picked up his fourth foul while guarding Westbrook.
Bryant scored the Lakers' first eight points of the second half to help his team weather the Thunder's charge back from a six-point deficit. Jeff Green had three dunks in the first 5½ minutes of the third quarter as Oklahoma City was able to run the fastbreak offense that had given L.A. trouble in the previous two games at the Ford Center.
Bryant had the answer, though, with a jumper and a 3-pointer to break ties and then another 3 that put the Lakers up 73-67. Just after that, he was whistled for his second foul in a 28-second span and headed to the bench.
The Thunder scored the next six points to tie it but then fell behind again, even with Bryant on the sidelines.
Luke Walton had a jumper and a 3-pointer on consecutive possessions to push the Lakers' lead to seven, and it was 91-84 after Artest beat the shot clock buzzer with a baseline jumper.