Braden (2-0) allowed four hits, walked four and struck out one. Braden was making his first start of the season, replacing Joe Blanton, who was traded to Philadelphia last Thursday, in the rotation. The left-hander made 14 starts last year, but threw less than five innings in each of his final five starts.
Hannahan put the Athletics ahead 3-1 in the fourth with his homer off Andy Sonnanstine (10-5). Hannahan is 3-for-8 with two homers against the Rays' right-hander.
Kurt Suzuki had an RBI double and Hannahan added a run-scoring single during a four-run eighth that made it 7-1.
The Athletics had scored a total of nine runs during the six-game skid.
Sonnanstine gave up three runs and five hits in seven innings. He has allowed four homers in 12 1-3 innings over his last two starts.
The Rays, who are in first place in the AL East, dropped to 39-16 at home this season. Tampa Bay has lost nine of 12 overall.
Tampa Bay took a 1-0 lead in the third on a bases-loaded sacrifice fly by Carlos Pena.
Oakland reliever Brad Ziegler didn't allow a run in the seventh and eighth to extend his career-starting scoreless streak to 23 2-3 innings, surpassing the AL record of 22 innings set by Boston's Boo Ferris in 1945, according to the Elias Sports Bureau. Since 1900, George McQuillan of the Philadelphia Phillies holds the major league record of 25 innings, which he set in 1907.
