The Rays overcame a four-run deficit, then let a two-run lead get away when the Orioles scored single runs in the eighth and ninth to forge a 7-7 tie.
Joyce ended it by launching a 3-1 pitch from Tommy Hunter, who had come on for his second inning of work.
Shelley Duncan highlighted Tampa Bay's comeback with a three-run homer during a four-run sixth and the Rays added three runs in the seventh with James Loney delivering a pinch-hit, RBI double in that frame.
Baltimore's Chris Davis went 4-for-4 and staked the Orioles to the early led when he hit a three-run homer in the first.
Exploding whale video goes viral on Internet
Puzzle-maker slips 'Murdoch Is Evil' into Rupert Murdoch's Sunday Telegraph