Ray Allen added 16 points and led a late-game charge for the Heat, which leads the best-of-seven series, 2-1. Game 4 is Monday night in Miami.
The Heat trailed by 15 in the first half and never led until early in the third quarter.
Paul George scored 17 points for Indiana, shooting five for 13 in his return after being concussed late in Game 2. Roy Hibbert scored 16 points, David West added 13 and Lance Stephenson had 10 for the Pacers.
Miami started the game two for 10 from the floor. The Heat then made 21 of its next 31 shots, including eight straight in the third quarter that gave the two-time defending NBA champion its first lead.
James' dunk with 7:36 remaining in the third put Miami up for the first time, 52-51. That was the first of nine lead changes in the quarter, before the Heat did what Coach Erik Spoelstra implored his team to do in a pregame locker room address.
"Impose our identity," Spoelstra said.
Eventually, the message reached the Heat, and its two best players led the way.
James' three-pointer with 1:21 left in the third put Miami up, 67-63, then its biggest lead of the night. Wade subbed in for James with 5.7 seconds left because the four-time most valuable player was dealing with what appeared to be a hamstring cramp and connected on a three-pointer with 1.4 ticks remaining for a 74-67 lead going into the fourth.
James retreated toward the locker room at that point, stopping halfway down the hallway known as "Championship Alley" while trainer Mike Mancias stretched him out. Meanwhile, Wade — not exactly known for behind-the-arc prowess — opened the fourth quarter with another three, the Heat was up 10 and the floodgates were opening.
Indiana got to 76-74 before Allen made a three-pointer, and with that, the tone was set for the final minutes. Allen made three three-pointers in the final 5:59, the last of those putting Miami up by 15.