Apart from a second quarter where Boston came alive with 35 points and Miami lost their intensity, the Heat and James had too much for the 17-times NBA champions.
It says much about James's current form that it did not feel like an outstanding display yet he produced 32 points on 13 for 22 shooting from the field for his sixth 30-point game in this season's play-offs.
It was by no means a classic, with the game lacking the intensity of the conference semi-finals both teams had to battle through, but Miami will take confidence from winning by 14 points while never really hitting their stride.
"We feel like we didn't play our best basketball," said James who also ended with 13 rebounds. "We had some great looks offensively but we had a lot of breakdowns defensively, so we didn't play our best.
"It's a good sign, a good thing for us but we will prepare and we will be better on Wednesday."
Boston, who did not secure their place in the conference final until Saturday's win over Philadelphia, lacked sharpness on offense and struggled at times with Miami's pace.
"They crushed us in the 50-50 game, anything loose or rolling, they got there," said Celtics coach Doc Rivers.
"Offensively we have to be more efficient. I didn't think we did great defensively but we still kept them to 93 points."
Boston also picked up five technical fouls, including one for Rivers for disputing a call which he described as "the worst I've ever had".
James outscored the Celtics by himself in the first quarter, making 13 points as Miami, seeking their second successive NBA finals appearance, established a 21-11 lead - the fewest points they had allowed in this season's play-offs.
But Boston dominated the second, drawing level by halftime helped by two three-point throws from Paul Pierce, Rajon Rondo finally getting into the paint and Ray Allen doing a good job keeping Miami's Dwyane Wade quiet.
Miami came out with determination after the break though, outscoring Boston 26-15 in the third quarter with James making 10 of those points to see the Heat to a 72-61 lead.
The Heat, solid in defense, were never in danger in the fourth quarter and were able to rest James and Dwyane Wade (22 points) in the latter stages.
"At times it was a strange game with some good runs from both teams," said Miami head coach Erik Spoelstra.
"We felt we could play better, I am sure they felt the same, but we found a way to grind it out, you have to against this team."