Paul Maholm (6-6) gave up one run on eight hits in six innings.
The Cubs have won two of the first three games in the four-game series after winning two of three in Chicago on May 7-9.
The Braves, third in the NL East, fell 6 games behind first-place Washington.
Rizzo's homer off Anthony Varvaro in the eighth was his second of the series and third overall since his recall from Triple-A Iowa on June 26.
Randall Delgado (4-9) allowed three runs on seven hits and two walks in six-plus innings. He has lost four straight decisions.
Michael Bourn led the Braves at the plate, racking up three hits and finishing a home run short of the cycle after recording a single, double and triple. Bourn scored Atlanta's only run on an RBI from Martin Prado.
Atlanta Braves first baseman Freddie Freeman fall into the dugout after catching a fly ball hit by Chicago Cubs' Alfonso Soriano in the eighth inning of a baseball game on Wednesday in Atlanta. / photo: Associated Press
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