The Rays loaded the bases with two out in the 10th and put two on with one away in the 12th without managing to push across the winning run.
Tampa Bay, however, struck quickly against Tom Wilhelmsen in the 14th for its second straight victory.
Wilhelmsen (3-2), working his second inning, struck out B.J. Upton to begin the 14th and then gave up a single to Pena. Zobrist followed with his double to right on a 1-0 pitch and Pena scored all the way from first.
The game had been deadlocked since the seventh inning.
Seattle managed just five hits off seven Tampa Bay pitchers.
Jake McGee (3-2) recorded the final two outs in the top of the 14th to get the win.
Carlos Peguero hit a two-run homer in the seventh to give Seattle a 3-2 lead, but Upton's infield single drove in the tying run in the bottom half of the frame. Upton hit a solo homer in the sixth.