The Rays extended the longest current winning streak in the majors to seven games.
It had been a 3-3 game since the sixth when Escobar reached on a one-out single in the 13th. Ryan Pressly (2-1) threw wildly on a pickoff attempt, allowing Escobar to reach second, and Luke Scott then drew a walk.
That brought up Zobrist, who drove a 1-2 pitch to deep center that was ruled a single and handed the Twins their fourth straight loss.
Minnesota built a 3-1 lead through four innings but Kelly Johnson tied the score in the Tampa Bay fifth with a two-run home run.
Cesar Ramos (2-2), one of 14 pitchers to appear in the game, retired the last two batters in the top of the 13th to get the win.