The Heat, who avoided playoff elimination in their last game on the road, returned home to clinch the best-of-seven Eastern Conference final 4-3.
The teams entered the final quarter tied but Miami, whose "Big Three" scored the team's closing 31 points, outscored the Celtics 18-6 in the game's final eight minutes.
Boston tried to slow down three-time league most valuable player James with frequent fouls, and he wound up with 12 free throws to go with nine field goals and 12 rebounds.
Bosh came off the bench to play a key role, especially in the second half. The seven-time All-Star's three-pointer, sandwiched around two James's field goals, was significant in a seven-point run by the Heat that gave the hosts an 88-82 lead.
Boston never came closer than four again.
Bass had 14 points in the first half to equal James's production, but got only two after intermission as the Heat cranked up their defense.
Fouls also hurt the Celtics. Kevin Garnett, who finished with 14 points, was saddled with his third foul with seven minutes to play in the opening quarter. Pierce also was in foul trouble.
Rondo again tried to carry the load in the second half, scoring nine consecutive Boston points at one point.
The NBA finals begin June 12 when Miami visit the Western Conference champion Oklahoma City Thunder.