Defending champions Cleveland scored a finals record 86 first-half points as they won 137-116, with the Warriors now leading the best-of-seven series 3-1.
Golden State were bidding for a perfect run of 16 wins from 16 in the play-offs - a feat never achieved before.
But Kyrie Irving and LeBron James shone as the Cavs forced a fifth game.
Irving scored 40 points while James added 31 to end Golden State's remarkable run of consecutive post-season victories. But the Warriors can still seal the championship at home on Monday (Tuesday, 02:00 BST).
"It was do-or-die. The job is still far from over," Irving said. "We're still in a hole. But this is a good start."
James added: "We have championship DNA as well. We showed that. We were in attack mode."
The Cavaliers won last year's series - also against Golden State - after trailing 3-1, and are now trying to become the first NBA team to recover from a 3-0 deficit.
Cleveland Cavaliers kept alive their NBA Finals hopes as they denied Golden State Warriors the win they needed for a historic series victory in game four.