Curry - the most valuable player in the NBA - hit seven three-point baskets to put the Warriors on the brink of winning successive titles.
No side has managed to overcome a 3-1 deficit to win the Finals and Golden State have home advantage for Monday's game five. "He's the MVP for a reason," said Warriors coach Steve Kerr of Curry.
"He doesn't have the size and strength to dominate a game physically so he has to dominate with his skill. He trusts his shot and he just kept firing and they went in," he added.
Curry had averaged 16 points in the first three games of the series and he scored 13 points in the fourth quarter on Friday.
In the final quarter, Cleveland went six minutes without scoring a basket and LeBron James tangled with Golden State's Draymond Green in the final three minutes of the game as tempers ran high.
"Draymond said something that I don't agree with," said James, who won the NBA Finals in 2012 and 2013 with Miami but has faced defeat in the last two, including last year's defeat by Golden State.
"I'm all cool with the competition, but some of the words that came out of his mouth were a little bit overboard. As a competitor, I love going against Draymond, and I'm all about going out there and leaving it out on the floor. But when it gets a little bit more than what it should be, that's what caused me to have words with him," he added.