Teixeira's RBI single in the top of the third inning started the Braves' scoring and his three-run homer in the top of the fifth was part of a five-run outburst as Atlanta built a 6-0 advantage.
Campillo, the winning pitcher, capped the fifth inning rally with a two-run single. Kelly Johnson added two hits, one a two-run single in the ninth.
Yunel Escobar had a run and three hits, while Chipper Jones had a run and two hits for the Braves, who had lost three of their previous four games.
Campillo (2-0) gave up an earned run and four hits in five innings with no walks and six strikeouts.
The losing pitcher was starter Seth McClung (2-2).