Boston lead by as many as 18 points in the first half but Miami almost clawed their way to an unlikely win on shot-making by LeBron James and smothering defence lead by Wade that took the Celtics out of their rhythm.
"Our execution in the first half was flawless, but we were unorganized the whole second half," said Boston coach Doc Rivers. "Miami got into our air space and took us out of everything. Their defensive energy took over."
James' three-point basket with 37 seconds to play in regulation sent the contest into overtime, before the league Most Valuable Player fouled out with about two minutes left in the extra session.
"In overtime, when you lose one of your big options ... we were just trying to survive," said Miami coach Erik Spoelstra.
"It was a heck of a (second) half, but it certainly wasn't a heck of a game. You have to be collectively willing to get into the pit and get your hands dirty. In the second half we really got into the grind, into the fight."
Paul Pierce top scored for Boston with 23 points before being benched in overtime after a sixth foul.
"It was like chess. They took our queen and then we took their queen," Boston guard Ray Allen said of the early departures of Pierce and James.
Veteran Celtics center Kevin Garnett added 17 points and 14 rebounds - 11 of them defensive.
"Kevin's our best defender. That's not a secret. He's our best communicator, he's our best defensive rebounder. He does it all," Celtics guard Rajon Rondo said of his team mate.
It was Rondo, though, who steadied the side after Pierce's exit, posting the team's final three points to finish with 15 for the game and 15 assists.
James led the Heat with 29 points. Wade had 20 points and chipped in with what Spoelstra termed "ferocious" defence.
Each team has now won twice on their home court. The action returns to Miami for Game Five on Tuesday.
"We look forward to going back home to our own fans," said James. "We've just got to come out with a sense of urgency ... this is what the post-season is all about."