The Warriors were down by 13 points during the first half but fought back in the third quarter.
The 4-3 win meant Golden State became only the 10th team to rally from a 3-1 deficit to win a play-off series.
The defending champions will now face a title rematch with the Cleveland Cavaliers and star player LeBron James.
"We survived by the skin of our teeth," said Golden State coach Steve Kerr. "We were able to pull it out, and we're moving on."
Warriors host the Cavaliers in the first game of the best-of-seven finals on Thursday, hoping to repeat their 4-2 success in the 2015 finals.
Steph Curry scored 36 points to lead the Golden State Warriors to a 96-88 victory over the Oklahoma City Thunder and into the NBA Finals.