The Rays quickly erased a 2-0 deficit and scored all their runs in the first four innings to win for just the third time in 11 games. The Rays were coming off a 2-7 road trip.
Tampa Bay batted around in the first against Brett Anderson with all four runs coming after two were out.
Evan Longoria contributed a two-run homer during a three-run fourth.
Alex Cobb (2-1) benefited from the offense, picking up the win by holding Oakland to three runs on 10 hits in 7 1/3.
Anderson (1-3) left the game after the first inning, having suffered what team officials said was a sprained ankle. Oakland has lost five times this season and three of those setbacks have come in games Anderson started.