James, who missed Miami's previous two games with an ankle sprain, scored 10 of his game-high 34 points in overtime but missed a potential go-ahead three-pointer in the final seconds as the Heat's woes continued.
The two-times reigning NBA Most Valuable Player also missed a three-point attempt in the dying seconds of regulation and was just 11-for-30 from the field in what was a night of poor shooting for both sides.
Joe Johnson forced overtime on a running jumper with 25 seconds left in the fourth and Josh Smith put Atlanta ahead for good to give the Hawks their seventh win in eight games.
Johnson ended with 19 points and 10 assists while Atlanta (28-15) also got 19 points from reserve Jamal Crawford.
"Guys knew this was going to be one of those games that even though it's January that you get measured on," Crawford said. "It's a good barometer."
Dwyane Wade recorded 27 points playing through a sore knee that kept him out of Monday's practice and Chris Bosh missed his first contest for Miami (30-13) with a sprained ankle he suffered during Saturday's loss to the Chicago Bulls.
Prior to their rash of injuries and recent slump, the Heat had won 21 of 22 games.
Miami led Atlanta by six with about three minutes left in regulation before the Hawks went on an 11-2 run. James answered with five straight points for the Heat as the battle waged into overtime.
The win moved Atlanta within two games of the Southeast Division lead held by Miami, which is second in the Eastern Conference.