Jeff Green's buzzer-beating three-pointer snatched the points for Boston, who in turn ended a three-match unbeaten run for Miami.
James top-scored for the Heat with 25 points, and provided more assists than any team-mate, but believes he could have provided even more if he had been fully fit.
"I am not 100 per cent," he conceded. "My back is not where I want it to be, but I can still be effective.
"I feel like I was effective tonight, but didn't make enough plays to help us win."
The defending NBA champions were two points ahead with 0.6 seconds of the game remaining before Green's late heroics.
The double MVP winner thought they should not have lost the game but praised Green for his match-winning shot.
"Obviously we can't afford to trade baskets with anyone, no matter who we are playing, and that is what we did tonight," he said.
"We covered as well as we could. It was a hell of a pass by Joe and a great catch and shoot. I was right on him. We messed up the game tonight man and it got us bad."
Heat coach Erik Spoelstra congratulated Boston on their victory – their third of the season – but bemoaned his side's inability to see the game out.
"Boston played well so give them the credit for taking this game tonight," he said. "They outplayed us, out-competed against us, but this game always has an energy and a karma about it.
"And we did too many things where we did not deserve to win this game and it cost us at the end. Even when it looked like we had daylight we were not doing enough of the little things or the tough things the way we are capable of.
"I think it is pretty clear we were just trying to win this game with our offence, and it is a shame that that type of offensive game got overshadowed by an extremely poor defensive effort."