Neither side had more than a two-point lead during the final nine minutes and the Warriors came up with the last points of the game when Barnes made two free throws.
During the closing minute the Clippers missed five field goal attempts, two of them layup tries by Chris Paul.
Andre Iguodala missed two free throw tries for the Warriors with nine seconds to play, failing in his bid to clinch the victory.
Paul then drove toward the basket in an attempt to force overtime, but his layup try was blocked out of bounds from behind by Klay Thompson with one second showing on the clock.
Jamal Crawford received the inbounds pass and put up a 3-point attempt at the buzzer, but it came up well short.
Paul scored 26 points and handed out 11 assists while Griffin collected 22 points before his ejection in the third period. He was kicked out of the game for picking up a second technical foul, which came as a result of a brief shoving match with Andrew Bogut.