Johnson scored 22 points and tied it at the end of regulation with a 3-pointer. That ended a dramatic fourth period during which the lead changed hands eight times.
Johnson then hit a jumper with 18 seconds left in the extra period that accounted for the final score and Atlanta survived when Kyrie Irving missed a jumper and Alonzo Gee was unable to convert a tip-in attempt in the final second.
Josh Smith scored 32 points with 17 rebounds for the Hawks, who won for the third time in four games. Atlanta swept the three-game season series with Cleveland and has defeated the Cavaliers seven straight times.
Irving scored 29 points to lead Cleveland, loser of four out of five.
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