The Spurs closed the game with an 8-1 run to win their fourth straight.
Stephen Curry scored 27 points and Jordan Poole added 15 for Golden State, which was playing a night after snapping Phoenix's 18-game winning streak in a matchup of the NBA's top teams.
White hit a go-ahead 3-pointer with 1:43 left in the game to give the Spurs a 107-106 lead.
"We didn't panic," White said. "They were up three and we just answered and found a way to win. It's not a good recipe for success. We want to keep the leads that we have, but just stay with it, compete and good things happened tonight.
"We knew they were going to go on a run. They're the Warriors, they've been doing it for years."
The Warriors trailed by as many as 22 early in the second quarter and by 18 in the third.
"We're thrilled because they are who they are and we could still get the win, although we know we caught a break," Spurs coach Gregg Popovich said.
Murray hit a 3-pointer with 8:15 left in the third that gave San Antonio an 80-62 lead.
The Warriors rallied for a 104-103 lead on Curry's step-back 3-pointer with 2:39 left that highlighted a 21-4 run.
Lonnie Walker IV scored 21 points coming off the bench and Keldon Johnson added 15 for the Spurs.
"The Spurs were great," Warriors coach Steve Kerr said. "They took it to us right from the outset. They came in playing well, they had a ton of energy, they ran right through us, right past us.
"We just had a tough time getting going, but the fact that our guys gave themselves a chance in the second half with that competitiveness was indicative of the kind of team we have and the kind of guys we have."
Damion Lee scored 14 points and Draymond Green had nine points, nine assists and seven rebounds for Golden State.
"They really showed their championship mettle by coming back and playing as hard as they did," Popovich said.
"You can see why it's a special program and why they're championship-caliber guys. It's not just about the talent, it's all the other stuff and they've got it in spades."
The Warriors had won 11 straight home games, tied for their longest home winning streak since they won 54 straight from Jan. 3, 2015, to March 29, 2016. All 11 wins on their most recent home winning streak were by double digits.
Spurs: At Phoenix Monday night. Phoenix beat the Spurs 115-111 on Nov. 22 in the only meeting between the teams this season.
Warriors: Host Orlando Monday night. Golden State hasn't lost to the Magic at home since Dec. 3, 2012.
White stops Warriors run with an and-1 right before half
Derrick White steps in front of the Draymond Green pass and gets the layup plus the foul.