The Rangers wasted an outstanding start by Derek Holland while losing for the second time in six games.
Mark Lowe (0-1) came on for his first inning of work in the 11th and opened the frame by issuing a walk to B.J. Upton on a 3-2 pitch. Zobrist then hit his 16th homer of the season.
That gave the win to Wade Davis (3-0), who struck out five while pitching the 10th and 11th innings.
Holland allowed only two hits and struck out 11 in eight innings. Tampa Bay starter Jeremy Hellickson gave up a run on four hits in six frames.