Longoria's run-scoring double in the third inning put the Rays up 2-0 and Escobar added an RBI single in the fourth before the game was delayed by rain for more than 2 hours.
After the break, Desmond Jennings added a pair of RBI doubles as Tampa Bay notched a third straight win and maintained a 1 1/2-game lead over Cleveland for the second and final American League wild-card spot.
The Rays lengthened their advantage over the New York Yankees to three games.
Brandon Gomes (1-1) came on after the rain delay to earn the victory with one scoreless inning after starter Matt Moore had tossed three scoreless innings.
Andrew Albers (2-3) was roughed up for four runs on six hits in a four inning start for the Twins, who dropped a fourth straight.