Los Angeles handed the Thunder their first loss of the playoffs and reduced its deficit in the Western Conference semifinals to 2-1. The Lakers will try to even the series when the teams meet Saturday in Los Angeles.
Oklahoma City rallied from an early 12-point deficit and the game turned into a tense battle throughout the second half.
A jumper by Kevin Durant gave the Thunder a 94-93 lead with 1:06 to play, but Los Angeles made the key plays down the stretch and was perfect from the foul line over the closing minute.
Bryant's two foul shots with 34 seconds left put the Lakers in front for good and World Peace then came up with a key steal against Durant that led to two more free throws with 12 seconds remaining.
After Kendrick Perkins hit a layup for the Thunder, Bryant was fouled with 10 seconds to play and his two foul shots accounted for the final score.
The Lakers swarmed around Durant as he attempted a long 3-point attempt that misfired just before the buzzer.
Bryant made all 18 of his foul shots and scored 36 points while Andrew Bynum had 15. Both Bynum and Paul Gasol grabbed 11 rebounds.
Durant paced the Thunder with 31 points. Russell Westbrook and James Harden had 21 each.