On the verge of being swept out of the playoffs, Wade erupted for 19 points in the fourth quarter where the Heat opened with a 25-8 run to cut the Celtics' best-of-seven series lead to 3-1.
No team has come back from a 3-0 best-of-seven deficit to win in the playoffs, but Miami at least forced the series back to Boston on Tuesday.
"We're always going to fight," Wade told reporters. "We fought today and I finally had a good shooting game the way I wanted to.
"I was going to shoot all those shots no matter what happened. I got hot at the right time."
Rajon Rondo had 23 points and nine assists while Kevin Garnett finished with 18 and 12 rebounds for the Celtics who had previously defeated the Heat in all six meetings this season.
Coming off Paul Pierce's buzzer-beater shot to deliver the win in Game Three, Boston fell behind by 13 after the first quarter but settled in to claim a six-point advantage after the third.
But that was when Miami's All-Star guard, who becomes a free agent at the end of the season, took over by making three quick three-pointers in the final quarter on his way to a playoff franchise-record scoring total.
James finished with 37 points, 12 rebounds and 11 assists to guide the Cavs to a 121-98 rout of the Bulls.
(Reporting by Jahmal Corner in Los Angeles, editing by Dave Thompson/Greg Stutchbury)