Anthony Davis scored 34 points for the Lakers, with James adding 25 points and nine assists.
Jimmy Butler had 23 points for the Heat, who lost point guard Goran Dragic and forward Bam Adebayo to injury.
James said: "The job is not done. We're not satisfied winning one game."
Speaking after Thursday's game, he added: "You have to get a feel for how hard Miami plays. We got a sense for that. We got more work to do. It's that simple."
The Lakers went into the series as overwhelming favourites to land a record-equalling 17th NBA Championship title, 10 years after their last finals appearance.
However, it was the Heat who began the game on top, racing into a 13-point lead in the first quarter.
That advantage would not last long, though, as the Lakers scored 11 three-pointers to complete a 30-point turnaround by the end of the first half - before surging clear in the second.
"They're a great team. They're here for a reason," said Lakers forward Davis. "They came out very hot, we came out slow.
"We can't come out like that in game two.
"If we want to win this series and become champions we've got to come out better from the start."
Miami's evening was made worse by Dragic and Adebayo both being forced out of the game.
Dragic, Miami's leading scorer in the post-season, suffered a foot injury in the second quarter, while Adebayo left with a shoulder strain in the third.
Butler was also forced to leave the court with a sprained ankle, but was able to return in the fourth quarter.
"I'll be OK. Some treatment and I get ready to go again. I've got to be ready to go," he said.
Game two of the best-of-seven series is on Friday (02:00 BST on Saturday).