Stromile Swift added 15 points and 13 rebounds for the Grizzlies, and Gay had 15 points.
The Hawks led 81-80 after three quarters. Miller scored 14 points in the third for the Grizzlies, who trailed by 17 points in the second period and 60-50 at the half. Memphis tied it at 76 on a dunk by Swift with 2:18 remaining in the third.
The Hawks built their lead to 88-80 on a free throw by Childress with 9:28 left.
Atlanta took its halftime intermission behind the scoring of Childress, who was 6-of-7 from the field, and Marvin Williams. Childress had 14 points and Williams 13.
"It was Marvin and Chills [Childress] early in the game, and I thought in the second half, Josh Smith really made an impact on the game, especially on the defensive end," Atlanta coach Mike Woodson said.
"He got some blocks and key rebounds, and he had the big putback and free throw at the end to put us over the top," he said.
Atlanta led 40-23 only 2:43 into the second quarter after an 11-0 run keyed by Salim Stoudamire, who scored five points. Pau Gasol and Swift had eight points apiece for the Grizzlies at the half.
Atlanta's leading scorer, G Joe Johnson (25.0) missed his second straight game with a right calf injury, as did G Speedy Claxton, with a sore left knee. ... Atlanta beat the Grizzlies in their only other meeting this season, 97-88, on Dec. 15 at Memphis.