Curry put the Warriors ahead at 102-98 in extra time with two pairs of free throws before Harrison Barnes made it 105-98 with a three-pointer at a raucous Oracle Arena in Oakland, and there was no way back from there for Cleveland.
"We were able to scrap tonight," Curry said courtside after the Warriors outscored the Cavaliers in shooting from the field by 44.3 percent to 41.5 with five of their players getting into double digits.
"The three-pointer in the corner (by Barnes) opened the game up and they scored just two points in overtime, that's huge."
The score was tied at 98-98 with 31.9 seconds left in regulation before Curry was denied a layup by a superb block from behind by Kyrie Irving, then LeBron James missed a jump shot for the Cavaliers with only 3.6 seconds remaining.
Curry had a team-high 26 points on 10-of-20 shooting while James weighed in with 44, his most ever in a Finals game, in a losing cause as the Cavaliers were outscored 10-2 in overtime.
Asked how his team had coped with the offensive threat posed by James, who is back in the NBA Finals for a fifth consecutive year, Curry replied: "These guys stick to the game plan. We know what he is about and what he's capable of.
"Our guys guarded him all night and made it tough on him. This is a long series."
James also added eight rebounds and six assists while big center Timofey Mozgov contributed 16 points for Cleveland.
Adding further concern for Cleveland, however, was the sight of their inspirational All-Star point guard Kyrie Irving limping off the court during overtime after an awkward fall.
Irving had been in dazzling form for his team at both ends of the court with 23 points, seven rebounds and four steals.
With James sparking the offense and Tristan Thompson spearheading a stifling defense, the Cavaliers initially outshot the Warriors from the floor and they streaked 14 points in front before ending the first quarter with a 29-19 lead.
Marreese Speights, off the bench, and Curry, who poured in a couple of threes, inspired a stirring fightback in the second quarter as Golden State twice moved five points ahead before Cleveland took a slender 51-48 advantage into halftime.
A running bank shot by Irving early in the third gave the Cavaliers a seven-point cushion but the Warriors, roared on by increasingly vocal fans, clawed back to end the quarter tied at 73-73, and set up a gripping finale.
Game Two is Sunday in Oakland.