Curry poured in 18 points in the final quarter as an out-of-sorts Warriors reeled in the Grizzlies with less than a minute remaining to close out victory and take a 3-1 lead in the best-of-seven series.
Memphis, who suffered a 142-112 mauling in game three on Saturday, had looked poised to punish one of the worst Golden State offensive displays in years to seize victory at San Francisco's Chase Center.
But after leading from the opening quarter until the final minute of the fourth quarter, the Grizzlies, without the injured Ja Morant, were unable to hold on.
And as Memphis tightened down the stretch, it was Curry who kept his cool to reel off a string of free throws that allowed Golden State to edge home.
Curry gave the Warriors their first lead of the night from the free throw line at 94-93 with 45.7 seconds remaining.
The eight-time All-Star then bagged six more free throws to help put the hosts five points clear at 100-95 with 9.7 seconds on the clock to effectively ice the game.
Andrew Wiggins finished with 17 points while Klay Thompson and Jordan Poole both had 14 points apiece.
Curry's points haul included a three-pointer that made him the first player to score 500 threes in the NBA playoffs.
But for long periods that statistical milestone looked like being of scant consolation for Golden State, who shot just 36 of 90 from the field all night.
Curry was at a loss to explain Golden State's lacklustre offensive display, which came so soon after Saturday's free-scoring game-three blitz.
"That's basketball," Curry said. "We got a little rushed in the first six minutes and I set the tone for some ugly offense.
"But credit to our defense. They kept us in it. We had a chance to weather the storm."
Golden State's task had been made more challenging by the absence of coach Steve Kerr, who tested positive for Covid-19 earlier Monday and was unable to take his seat on the bench.
Jaren Jackson Jr. led the Memphis scorers with 21 points with Tyus Jones adding 19.
Memphis coach Taylor Jenkins was left ruing his team's agonising late loss.
"We played well for 48 minutes, the ball just didn't bounce our way a couple of times late in the fourth quarter," Jenkins said. "We've just got to find a way to play better and win game five."
- Horford sinks Bucks -
In Milwaukee, Al Horford scored 30 points as the Boston Celtics produced a devastating late scoring burst to defeat the Milwaukee Bucks 116-108 and knot their best-of-seven series at 2-2.
The defending champion Bucks looked to be heading towards a 3-1 lead in the Eastern Conference semi-final series after stretching into an 11-point lead late in the third quarter, with Giannis Antetokounmpo in commanding form.
But a devastating fourth-quarter shooting display from the Celtics -- with Horford scoring 16 points as Boston outscored Milwaukee 43-28 -- transformed the contest as the Celtics head home to Boston for game five on Wednesday.
Horford's total was a career-high in playoff games and gives the Celtics a crucial advantage with two of the three remaining games in the series taking place in Boston.
Horford received scoring support from Jayson Tatum, who added 30 points, while Jaylen Brown and Marcus Smart both finished with 18 points apiece.
Derrick White added 11 points from the bench as five Celtics players finished in double figures to silence Milwaukee's Fiserv Forum crowd.
"Our backs were against the wall. We needed to respond and that's what we did," Horford said.
"We didn’t quit, we just kept pushing," the 35-year-old five-time All-Star added.
"Playoffs is hard. You have to grind and get down to it... we just needed to find a way."
Antetokounmpo led the scoring for Milwaukee with 34 points in a losing effort, with 18 rebounds and five assists, while Brook Lopez finished with 17 points.
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