James came on late after a slow start in which he missed his first 10 shots. He was favoring his right leg for most of the fourth quarter, but hit a three-pointer and a runner late in overtime to seal the win. The Hawks were able to battle with Cleveland despite being without Kyle Korver due to injury, and Al Horford after he was ejected in the first half.
Shelvin Mack missed two three-pointers in the final seconds, as the Hawks attempted to force a second overtime.
Jeff Teague led Atlanta with 30 points, while Paul Millsap added 22 for the short-handed No. 1 seed in the east. They came back from a 10-point deficit in the fourth quarter, but now trail the series 3-0, a deficit that no team in NBA history has been able to overcome.
The Cavs got out to the 10-point lead in the third, where James would record 15 points, six rebounds and six assists.
Cleveland also played shorthanded, as Kyrie Irving missed his second straight game with an injured left knee. They would get contributions from the likes of Matthew Dellavedova and J.R. Smith, who each scored 17 points.
The Cavaliers will look to advance to their first NBA Finals since 2009, during LeBron's first stint with the franchise. It would be the Cavs' second Finals appearance, and LeBron's fifth in a row. Game 4 of the series will take place in Cleveland on Tuesday night.