The visitors outscored the Warriors by 24 points down the stretch en route to a 120-108 win to take the early advantage in the best-of-seven series.
The result stands in contrast to an electric first quarter for Golden State, buoyed by a blazing opening 12 minutes from Steph Curry.
The sharpshooter knocked down six threes in the first to guide the Warriors to a 32-28 lead.
However, Boston’s defence adjusted in the break and Curry cooled off from his hot start, allowing the Celtics to dictate the pace of play and grind to a 56-54 half-time lead.
Just as they had done all season, the Warriors exploded in the third on the back of Curry’s seventh three, at one stage building a 14-point lead before closing the quarter with a 12-point buffer.
But they were unable to carry that momentum into the fourth, with Jaylen Brown sparking a game-changing 17-0 run with 10 points in the first five minutes.
Brown finished with 24 points for the Celtics, while Curry led all scorers with 34.