With the score tied 3-3 in the seventh, Escobar led off the frame with a go-ahead solo shot against Twins starter Samuel Deduno.
Desmond Jennings followed with a triple, chasing Deduno, and Caleb Thielbar took the mound only to surrender a two-run blast to Zobrist.
Luke Scott also went deep for the Rays, who reeled off a fifth straight win and improved to 12-3 in their last 15 games.
Alex Torres (3-0) was credited with the decision with a scoreless seventh inning after Rays starter Roberto Hernandez had yielded three runs on eight hits over six frames.
Deduno (4-4) ended up allowing five runs on 10 hits over six innings for fading Minnesota, which fell for the eighth time in nine games.
Aaron Hicks went 4-for-4 with two runs scored and Justin Morneau homered as the Twins dropped the first of a four-game series.
Celebrity Families of 2014 [PHOTOS]