Tigers starter Justin Verlander (0-1) carried a 2-0 lead into the ninth but couldn't hold it.
Evan Longoria followed with a game-tying single.
Reliever Daniel Schlereth issued a walk to pinch-hitter Elliot Johnson, then gave way to Jose Valverde, whose 3-2 delivery to Zobrist was grounded up the middle for the margin of victory.
The comeback made a winner of James Shields (1-0), who yielded two runs on six hits over eight innings, walking one and striking out five.
Fernando Rodney tossed a scoreless ninth for his third save as the Rays evened their three-game series with the Tigers.
Miguel Cabrera went 2-for-4 with an RBI single for Detroit, which endured its first loss of the season after opening with four straight victories.