The two-time NBA Most Valuable Player scored 22 of his 36 points in the fourth quarter and overtime, with key three-pointers and a timely block helping Golden State stretch a home win streak to seven games.
"We feed off our crowd's energy," Curry said. "Whatever it is in terms of our focus and energy and effort, it's giving us a chance no matter what the score is down the stretch."
The Bucks, playing without two-time NBA MVP Giannis Antetokounmpo due to his sore right hand, saw their nine-game road win streak snapped and their NBA-best record fall to 48-19.
Milwaukee led by five in the final minute, but Curry sank a three-pointer, made a fast break layup to tie the game, then answered a Jrue Holiday three with one of his own with 18 seconds remaining to lift the Warriors level at 111-111.
His late block of Holiday in the paint on a drive to the hoop helped ensure the game went into overtime.
"It felt like playoff-type energy and atmosphere," Curry said. "I know they were missing Giannis but we've been searching for a long time to try and figure out how to perform and execute down the stretch in a game like this.
"We got down five with under a minute and come back. And I think it has been a long time since we won an overtime game. So all those things -- dig deep and figure out a way to get it done."
Curry, who turns 35 on Tuesday, scored nine points in overtime to seal the victory, playing 43 minutes in all to achieve his first victory in four tries since returning March 5 from a left leg injury.
"I'm great. I'm just getting old," a visibly worn-down Curry said after the finish. "Everybody contributed and that's the way it has to be for us the way we're built right now."
Curry made eight of 11 shots from the floor in the fourth quarter and overtime, and sank four of seven from behind the arc.
"That's a great way to get a win," Curry said.
In Atlanta, Jason Tatum delivered game highs of 34 points and 15 rebounds in powering the Boston Celtics over the host Hawks 134-125.
Jaylen Brown added 24 points as the Celtics improved to 47-21, while the Hawks, paced by 35 points and 13 assists from Trae Young, fell to 34-34.
- Kawhi sparks Clippers -
Kawhi Leonard scored 38 points to lead the Los Angeles Clippers over the visiting New York Knicks 106-95.
Paul George added 22 points, eight rebounds and eight assists for the Clippers. Immanuel Quickley led the Knicks with 26 points.
Clippers guard Russell Westbrook overtook Isiah Thomas for ninth on the NBA's all-time assists list with a third-quarter pass to set up a George dunk for his 9,062nd career assist.
Talen Horton-Tucker had game highs of 37 points and 10 assists in leading Utah to a 119-111 victory at Charlotte.
Zach LaVine scored 36 points and DeMar DeRozan added 27 to lead the Chicago Bulls to a 119-111 victory at Houston.
Wendell Carter scored seven of his 27 points in overtime to spark Orlando over visiting Miami 126-114. Jimmy Butler scored 38 points to lead the Heat.
Shai Gilgeous-Alexander scored 35 points to lead Oklahoma City's 110-96 victory at New Orleans.
Desmond Bane scored 25 points while David Roddy came off the bench to score 24 and spark the Memphis Grizzlies over visiting Dallas 112-108.
Tim Hardaway Jr. led the Mavericks with 23 points, but Dallas was without Luka Doncic due to a left thigh strain and Kyrie Irving with a right foot injury, while Ja Morant was still away from Memphis for personal reasons.
Harrison Barnes scored 19 points to lead seven double-digit scorers for Sacramento in a 128-119 victory at Phoenix.
Jalen Smith scored 20 points off the bench to lead Indiana's 121-115 victory at Detroit.