Miami embarked on a five-game road trip and found things difficult against a team trying to fight its way out of the Atlantic Division cellar.
The 76ers battled back from a 12-point halftime deficit and tied things up with 1:20 to go when Jrue Holiday scored on a dunk.
James then put Miami in front for good with a free throw and Wade scored a crucial basket on a tip-in of a James miss with 29 seconds remaining.
Two foul shots by Wade boosted the Miami lead to five points with 22 seconds to play and the Heat went on to extend the NBA's longest winning streak this year.
James scored 27 points to go with seven rebounds and eight assists while Wade contributed 21 points. Miami made 23 free throws to just seven for Philadelphia.
Miami's trip will continue with stops in Milwaukee, Toronto, Boston and Cleveland.
The 76ers were led by 24 points from Thaddeus Young and 21 from Holiday.
Miami tied idle San Antonio for the most victories in the league with 49.