In another NBA play-off game, the Chicago Bulls beat the Indiana Pacers 88-84, while the Dallas Mavericks were stung by the Portland Trail Blazers 97-92 at the Rose Garden.
The Heat and Bulls seized 3-0 leads in their best-of-seven series with their victories away from their home courts, while Portland cut the Dallas's series advantage to 2-1.
LeBron James hit a three-pointer on the Heat's first possession of the fourth quarter to spark a 15-4 run and turn a two-point deficit into an insurmountable nine-point Miami lead.
"It was a gutsy win," Heat coach Erik Spoelstra said. "We could not really take control of the game until the fourth quarter. It was not pretty. It was not clean.
"But we found a way and that's the most important thing, just to gut it out and find a way. Now we move on."
Dwyane Wade led the Heat with 32 points and 10 rebounds, James had a double-double with 24 points and 15 rebounds and Chris Bosh scored 19 points.
Elton Brand led the 76ers with 21 points and 11 rebounds.
In Chicago, Derrick Rose scored his only field goal of the second half on a driving lay-up with 17 seconds left in the fourth quarter to push the Bulls ahead 86-84.
Ronnie Brewer hit two free throws with one second remaining to stretch the Bulls' final lead to four and ice the victory.
Both teams led the back and forth game by as many as five points in the final period, which was in doubt until the final second.
Rose was just 4-for-18 from the field but connected on 13 of 15 free throw attempts to lead the Bulls with 23 points.
Luol Deng added 21 for the Bulls, while Danny Granger led the Pacers with 21 points.
Portland broke a 72-72 tie with a 15-2 run straddling the end of the third and start of the fourth quarters and then held off the Mavericks rally.
Dallas narrowed the Portland lead to justy three points with 10 seconds remaining.