The 24-year-old Lewis became the first player in Twins history to hit grand slams in back-to-back games. He has three grand slams among his first 10 career home runs.
Jorge Polanco and Matt Wallner also homered for Minnesota, which has won four of its past five games. The Twins increased their American League Central division lead to seven games over the second-place Guardians.
Bo Naylor homered for Cleveland, which has lost four of six and eight of 12.
Twins left-hander Kody Funderburk (1-0) earned the victory in his major league debut. The 26-year old, who was the Twins' 15th-round pick in 2018, pitched two scoreless innings of relief and struck out three.
Guardians right-hander Xzavion Curry (3-3) allowed a season-high six runs on six hits in two innings. He walked one and struck out two.
Cleveland's hitters provided Curry with a 4-0 lead in the top of the second before the Twins surged back. Will Brennan hit a sacrifice fly, Gabriel Arias followed with an RBI double, and Naylor blasted a two-run homer to center.
Minnesota answered with six runs in the bottom of the second. Carlos Correa scored the Twins' first run on a balk, and Donovan Solano made it 4-2 with a run-scoring infield single.
Three batters later, Lewis put the Twins on top for good. He stepped to the plate with two outs and the bases loaded and drove a curveball over the wall in left-center field.
Andres Gimenez pulled the Guardians within 6-5 when he drove in Kole Calhoun on a fielder's choice groundout in the third.
Wallner made it 7-5 for the Twins by crushing a 438-foot homer to right-center field in the bottom of the third.
An RBI double by Steven Kwan trimmed Minnesota's lead to 7-6 in the fourth.
Polanco capped the scoring with a three-run homer in the bottom of the frame. His 437-foot shot to left field marked his 10th home run.
—Field Level Media