Joel Peralta took the mound and Jackson greeted him by drilling a pitch off the left-center field wall, handing the A's the lead.
Cliff Pennington added a two-run homer for Oakland, which took the opener of a four-game series against the Rays.
Brian Fuentes (2-8) collected the win after getting the final two outs of the seventh and Andrew Bailey racked up his 12th save.
Ben Zobrist went 3-for-4 with a solo homer and Kelly Shoppach added a bases-empty shot for Tampa Bay.
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