Curry, a four-time NBA champion who missed 27 games from February to early April with a nagging knee injury, shook off a slow first half to finish with 35 points.
He connected on seven of 12 three-point attempts to help the Warriors, who trailed most of the way, battle back from a 13-point fourth-quarter deficit.
His last three-pointer put the Warriors up 120-117, and they held on to knock Kawhi Leonard and the Clippers out of the post-season.
The Warriors, who finished the regular season 10th in the Western Conference, will have to win one more game to reach the play-offs proper when they face the Phoenix Suns on Friday.
The winner of that game will secure the Western Conference eighth seed and a first-round series against the top-seeded reigning champion Oklahoma City Thunder.
Curry said the key in Phoenix will once again be "resiliency."
"These win or go home scenarios, we think we know the team but you've got to be able to withstand the runs and stay in it emotionally.
"It was back and forth the whole game, and then we got over the hump the last three minutes."
Gui Santos and Kristaps Porzingis scored 20 points apiece for the Warriors and Al Horford made four three-pointers in the fourth quarter.
Bennedict Mathurin scored 23 points to lead the Clippers. Leonard and Darius Garland scored 21 points each, but Leonard was held to just two points in the final period.