Dwyane Wade added 22 points and seven assists for the Heat, who never trailed and led by as many as 17.
Miami host game two on Tuesday, with the winners of the best-of-seven series advancing to the NBA Finals to face either Oklahoma City or San Antonio.
Shane Battier scored 10 points, Mario Chalmers had nine and Mike Miller finished with eight as the Heat again coped admirably without star forward Chris Bosh.
Bosh remained remained sidelined with a strained abdominal muscle, and Heat coach Erik Spoelstra said there was no timetable for his return.
Kevin Garnett scored 23 points and added 10 rebounds for Boston, but the Celtics looked a step slow most of the night after wrapping up a seven-game second-round victory over the Philadelphia 76ers on Saturday night.
The Heat out-rebounded the Celtics 48-33, and blocked 11 Boston shots.
Rajon Rondo, whose triple-double led the Celtics in game seven against the Sixers, finished with 16 points, nine rebounds and seven assists. Paul Pierce scored 12 points, connecting on just five of 18 shots from the floor.
The Celtics started firing in the second quarter, battling back from an 11-point deficit to tie the score at halftime.
But the Heat, led by NBA Most Valuable Player James and Wade, inexorably pulled away in the third.
"The second quarter we executed our stuff offensively," said Celtics coach Doc Rivers. "Then, third quarter, the game just got away from us. We rushed. Our offense was rushed. We never got into our rhythm."