San Antonio could find out as soon as Monday night who it will play next. Oklahoma City leads the Lakers 3-1 in their series, with Game 5 on Monday.
Chris Paul had 23 points and 11 assists, Blake Griffin added 21 points, and Eric Bledsoe had 17 for the Clippers, who blew a six-point lead in the fourth quarter.
Both nursing injuries, Griffin and Paul combined to score 21 of the Clippers' 28 points in the third quarter when they took their first lead of the game. In the fourth, Bledsoe came up big, scoring 11 in a row, to extend the lead to 90-85 with 5:38 remaining.
After Reggie Evans missed two free throws, Green hit a 3-pointer to tie it at 92.
Duncan's hook put the Spurs in front 96-94 before Paul tossed up a one-handed shot that rose to the top of the backboard before dropping straight in as he crashed into a baseline photographer. He made the free throw to put the Clippers ahead by one.
Parker scored consecutive baskets and the Spurs regained the lead, 100-97, with 1:47 left.
Paul's two free throws drew the Clippers within one. After a timeout, Paul drove the basket and lost the ball. He fouled Green, who made the first and missed the second to keep the Spurs ahead 101-99.
Paul then missed another shot, and Mo Williams fouled Parker, who missed the first and made the second with a second left.
DeAndre Jordan added 10 points. Paul had best performance of the series after sub-par efforts in the first two games. He'd been playing with a strained right hip, while Griffin has a sprained right knee, an injured left hip and got stitches for a cut lip in the first half.
Neither team led by more than five points in the third period. Duncan had 10 points for the Spurs. Bledsoe's putback slam dunk gave the Clippers 75-74 lead going into the fourth.
The Spurs stretched their lead to 12 points with Duncan sitting out the opening 7 minutes of the second quarter.
The Clippers closed on a 14-6 run to trail 51-47 at halftime. Paul got it started with a 19-foot jumper and ended it with a 3-pointer. Early in the spurt, Griffin ran into Ginobili's shoulder, fell and one of his top teeth went through his lip. He left the court with 2:20 remaining to get two stitches on the inside and outside of his lip.
Los Angeles began the game on a 9-4 run before San Antonio scored 14 unanswered points, reminiscent of its 24-0 third-quarter spurt Saturday that led to the Spurs' eventual 10-point victory in Game 3.