Chris Bosh scored 19 points for the Heat, who rallied from an early 10-point deficit to move within one win of a sweep of the Eastern Conference series.
Game 4 is Sunday in Philadelphia.
James hit the court hard late in the fourth quarter and briefly sat against the scorer's table. He grimaced when his teammates pulled him up and he slowly walked back to Miami's bench.
The Sixers were determined to win one on their home court and played like a team trying all it could to get the series back to Miami.
Elton Brand had 21 points and 11 rebounds for the 76ers.
Wade missed practice before Game 2 with a migraine headache. Hours before Game 3, Wade said he was fine.
"I don't have to worry about bright lights," he said.
He was fantastic under the postseason spotlight.
Wade delivered with a string of big baskets - he was 10 of 19 from the field overall - in the third and fourth quarters that spurred the Heat to the brink of the conference semifinals. He drew a foul on a spin move against Jrue Holiday and hit two free throws with 51.6 seconds left that about clinched the game.
"We're a desperate team right now," James said. "We want to continue to play that way. D-Wade had an unbelievable game ... and we're ready to close it out."
Wade was 12 for 12 from the free throw line. For all the hype around the Big Three, Wade remembers he was the Biggest One in Miami. He still plays that way and the Heat needed all of his effort against the scrappy Sixers.
Lou Williams missed a 3 in the final minute that would have made it a one-point game. Holiday scored 20 points and Williams finished with 15.
For most of the game, it seemed like a Game 1 rerun. The Sixers' fast start would get picked away by Miami, the Big Three would take over and the Heat would roll.
When James hit a pull-up jumper for a 54-52 lead early in the third for Miami's first lead of the game, the plan was going as scripted.
The Sixers had a whole new ending planned.
Jodie Meeks and Holiday bookended 3s during a stunning 12-0 run that helped them regain the lead in front of rollicking crowd that truly made it feel like big-time, meaningful basketball was alive and thriving in Philadelphia.
In a play that showed what make the Heat great, James stumbled and flipped an underhand pass to Wade for a ferocious dunk that cut it 68-64.
"It definitely started a run for us," James said. "Luckily I was able to get rid of the ball before I traveled and D-Wade was able to slam it home. It's a great play for us, a great team win and we're looking to close them out on Sunday."
King James played jester when he fired an airball late in the quarter - the crowd reveled in that miscue - and the Sixers kept a 75-73 lead into the fourth.
Fans booed each time James touched the ball, but they sure got a kick out of his one misfire.
Philadelphia's fast start fizzled down the stretch and a team that showed amazing resiliency all season simply doesn't have the talent to match up with the Heat.
The 76ers made their first four shots, raced to a 9-0 lead and drove a rare pumped sellout crowd bonkers.
Philadelphia's stellar first quarter was a familiar feeling for both teams. In Game 1, the Sixers made eight of their first 10 shots and led 25-11. While the Sixers coughed up the lead in the second quarter in that one, they stretched the lead to double figures and never lost the lead, going into halftime up 52-50.
Brand, who scored only 20 points the first two games, established his mid-range jumper and scored 13 points in the first half. The Sixers only turned the ball over twice and proved the Game 2 flop was well behind them.
Williams delivered on his two pregame promises. The first compared Philly's fans to Miami's: "Ours will be better."
Fueled by "Rocky" clips, 20,404 fans were stoked from the time the lights were dimmed for intros to the final horn.
The other: The 76ers would "fight hard, fight to the finish." They did. They just didn't win.
NOTES: Former New York Giant Michael Strahan was at the game. ... The Heat don't care about fans' expectations. "We're not trying to prove anything. We're just trying to win," Bosh said.