Kendry Morales led off the inning against Grant Balfour (0-1) with a single and moved to second on a sacrifice bunt by Reggie Willits. He took third on Balfour's wild pitch and came home with the game-winning run on Rivera's medium-deep fly ball to center.
Trevor Bell (1-0) got credit for the win after relieving Jason Bulger with two gone in the top of the 11th and recording the final out.
Mike Napoli connected for a two-run homer and Torii Hunter added a solo shot and drove in two in the Angels' win.
Los Angeles took a 3-1 advantage into the ninth but Brian Fuentes surrendered the lead when Willie Aybar smacked a solo homer and Evan Longoria delivered a two-out, two-run double to force extra innings.
John Jaso went 3-for-5 with an RBI in Tampa Bay's loss.