LeBron James scored a playoff career-high 51 points in a losing cause as the Warriors took a 1-0 lead in the best-of-seven championship series, which continues Sunday in Oakland.
Golden State's Klay Thompson added 24 points and Draymond Green contributed 13 points, 11 rebounds, nine assists, five steals and three blocked shots.
Green taunted Cleveland's Tristan Thompson to spark a shoving incident in the final seconds of overtime that figures to carry over in the emotional fourth consecutive finals between the Cavaliers and Warriors.
Golden State won the 2015 and 2017 crowns. The Cavs took the 2016 title.
Four-time NBA Most Valuable Player James scored seven points in a 10-2 run that gave Cleveland a 104-102 lead with 50.8 seconds remaining -- a layup, dunk and 3-point play by James plus a Love 3-pointer.
Durant was whistled for a charging call but it was overturned on video review into a blocking foul on James and Durant equalized with two free throws at 104-104.
James answered inside but Curry drove for a layup, got fouled by Love and sank the free throw for a 107-106 Golden State lead with 23.5 seconds in regulation time.
George Hill was fouled and made 1-of-2 free throws with 4.7 seconds remaining but missed the second and the game went to overtime.
Golden State dominated the extra five minutes, starting it with a 9-0 run. Klay Thompson hit two 3-pointers and Durant another to seal Cleveland's fate.
In the final seconds, Green waved bye to Tristan Thompson after a foul and had the ball shoved in his face as players gathered to keep the two apart as Thompson was ejected and tempers were tested.
The Warriors seek their third title in four seasons and sixth overall.
James is playing in his eighth consecutive finals and ninth overall seeking his fourth career title and the second for the Cavs.
Physical intensity was on display from the start, with Cleveland's J.R. Smith slipping and falling into Klay Thompson, forcing the Warriors guard to the locker room for a spell in the first quarter with a bruised left leg.
James drove to the hoop in the second quarter only to have Green deliver a left hand to the face that sent James to the floor clutching his face. Green was whistled for a technical foul.
The Warriors closed the first half on a 16-5 run and began the third quarter with a 10-3 spurt for a 66-59 edge.
But James scored seven consecutive points, capping the run with a 3-pointer, to pull the Cavs level at 68-68 with 5:27 remaining in the third, James hitting 11 of his first 13 shots.
Golden State answered moments later with a 14-4 run, Curry and Thompson sinking 3-pointers and Durant's six points including a slam dunk to cap the spurt as the Warriors led 84-78 entering the fourth quarter.
Curry sank a 38-foot running 3-pointer at the half-time buzzer to lift the Warriors level at 56-56 after Cleveland had led by as much as 49-40 in the second quarter.
James had 24 points in the first half while Curry led the Warriors with 18.