Paul, who strained his left hamstring in the first quarter, scored over Spurs forward Tim Duncan to seal the win and finished with 27 points as the Clippers advanced to the conference semi-finals against the Houston Rockets.
Game One is on Monday in Houston.
Clippers forward Blake Griffin added 24 points, 13 rebounds and 10 assists for his second triple-double of the series.
Duncan finished with 27 points and 11 rebounds to lead the Spurs. Point guard Tony Parker had 20 points.
Griffin’s two free throws with 3:35 remaining put Los Angeles up 102-101 but Duncan’s putback and forward Kawhi Leonard’s follow-up from a miss by Patty Mills lifted the Spurs to a 105-102 lead.
A three-pointer from Barnes tied the score again before Parker’s putback made it 107-105 for the Spurs with 1:23 remaining.
Clippers guard Jamal Crawford scored on a drive to tie the score and Paul’s two free throws with 13.3 seconds left gave the Clippers a 109-107 lead.
Duncan converted two foul shots with 8.8 seconds remaining to knot the score at 109-109 before Paul’s last-gasp winning basket.