Cavaliers forward LeBron James had a chance to win it in regulation, but his shot in the lane fell short. He claimed he was fouled, but there was no whistle.
James led all scorers with 21 points, seven rebounds and six assists in a game played at Nationwide Arena.
Johnson finished with 17 points, including nine in the extra period, and 10 assists.
Hawks point guard Tyronn Lue added 13 points, while Zaza Pachulia added 12 and Josh Smith and Josh Childress had 10.
Drew Gooden contributed 17 points and 11 rebounds for the Cavaliers. Donyell Marshall scored 15 points and Zydrunas Ilgauskas had 12 points, 14 rebounds and four blocks.
The Hawks took a 37-28 lead at halftime. The Cavaliers started the second quarter with four consecutive turnovers, missed their first nine shots and were outscored 11-0.