Thome's bat had been about the only thing missing so far this season for the Indians, 8-1 after winning the first three games of their four-game set with Minnesota.
Thome came in batting just .143 and hadn't hit a homer before connecting in the second inning off Rick Reed (1-1), who never should have had to face Thome.
Last year, Thome batted .182 with three homers and 10 RBIs in the season's first month, but .307 with 46 homers and 114 RBIs the rest of the way.
Thome had two chances to hit a second slam, but struck out with the bases loaded in the sixth and eighth.
Dustan Mohr homered and Torii Hunter had three hits for the Twins, who never recovered after shortstop Cristian Guzman made two errors in the second, leading to six unearned runs.
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Danys Baez (2-0), a middle reliever last season but currently in Cleveland's rotation, gave three runs and six hits in 5 1-3 innings as Indians starters improved to 8-0 with a 3.04 ERA in nine games.
At the same juncture last season, Cleveland starters were 2-3 with a 6.89 ERA.
And Cleveland's offense, which was supposed to have trouble scoring this season, is averaging an AL-best 6.7 runs per game.
Baez was pulled after the Twins put two on in the sixth, but Mark Wohlers came on and struck out Corey Koskie and Mohr -- both looking -- to end the threat.
Wohlers, Jerrod Riggan and Charles Nagy combined for scoreless relief over the final 3 2-3 innings.
Reed should have gotten out of the second inning unscathed, but Guzman, who led AL shortstops with 21 errors last season, made two in the span of three batters as the Indians scored six unearned runs to take a 7-2 lead.
With one out, Guzman lazily backhanded Ricky Gutierrez's grounder but had it roll up his glove for an error. After Milton Bradley singled, Einar Diaz hit a routine double-play ball at Guzman, who let it go through his legs and Gutierrez scored.
Matt Lawton followed with an RBI single, and after Ellis Burks walked with two outs to load the bases, Thome officially ended his early season slump with his sixth career grand slam.
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Thome, who came in with only one extra-base hit, gave the Indians a 1-0 lead in the first with an RBI double.
Two walks got Baez in trouble in the second and the Twins took a 2-1 lead on Mohr's RBI single and A.J. Pierzynski's sacrifice fly.
Reed allowed two earned runs and six hits in five innings.
Notes: Bradley appeared to pull a muscle while running the bases in the eighth and left the game. ... Cleveland's run in the eighth snapped an 11-inning scoreless streak for Minnesota's bullpen. ... When Indians SS Omar Vizquel, who has won nine straight Gold Gloves, got his new glove from Rawlings -- which presents the Gold Glove Awards -- his name was spelled incorrectly as "Visquel." ... Guzman has made 67 errors the past three seasons. ... Twins OF Jacque Jones extended his hitting streak to a career-high nine games with a leadoff single in the first. ... Mohr's homer was just the second allowed by Cleveland pitchers this season.
