Trailing 1-0 entering the frame, the Twins sent 13 men to the plate, banging out seven hits, drawing a pair of walks and taking advantage of a Cleveland error that made seven of the 10 runs unearned.
Morneau added a second homer in the fourth -- the 200th of his career -- finishing 3-for-4 with four RBI, and Joe Mauer drove in three runs on a trio of hits as the Twins claimed the opener of a three-game series.
Ben Revere extended his 20-game hitting streak, going 4-for-5 with an RBI, as Minnesota recorded its fourth win in five games.
Scott Diamond (10-5) yielded three runs (two earned) on seven hits over seven frames to collect the victory. The left-hander walked one and fanned three.
Indians starter Zach McAllister (4-4) was bombed for nine runs (two earned) on six hits in 1 2/3 innings.
Carlos Santana socked a two-homer for the reeling Tribe, who lost their 10th straight.