Alexi Casilla started the ninth-rally with one out, lining a pitch from Rafael Perez (1-2) down the left-field line for a ground-rule double. After Mike Redmond struck out, Morneau doubled to left-center, scoring Casilla for a 3-2 lead.
Jason Kubel drove in Morneau with a single to give Minnesota a two-run lead.
Minnesota took two of three in the series to improve to 29-18 in the AL Central entering a four-game series at home that starts Monday night against the first-place Chicago White Sox. The Twins pulled within 2 1/2 games of Chicago, which lost 6-4 to Detroit.
The last time the teams met, the White Sox swept four straight by a combined score of 40-15 in Chicago. Minnesota has gone 26-14 since that series.
The Indians have lost 4-of-5 and are 13 games under .500.
Cleveland took a 1-0 lead in the second on a sacrifice fly by Sal Fasano.
The Twins loaded the bases on a walk and a bunt single, and then scored their runs on a fielder's choice grounder by Denard Span and a groundout by Casilla.
Sowers, coming off his first win in more than a year July 19 at Seattle, gave up two runs and three hits over eight innings, striking out five.
Twins starter Nick Blackburn allowed one run and four hits over seven innings and left with a 2-1 lead. The rookie right-hander got his first career win against the Indians on April 19 and is 2-0 with a 1.33 ERA in five games -- including three starts -- against them.
The Indians tied the game in the eighth off reliever Dennys Reyes on a sacrifice fly by David Dellucci that was nearly an extra-base hit. Center fielder Span raced over and made a lunging catch in left-center.
Sizemore got his 25th stolen base to go with 25 homers, joining former Indians OFs Bobby Bonds and Joe Carter as the only Cleveland players to do it. ... Twins LHP Craig Breslow, claimed off waivers from Cleveland on May 29, hasn't allowed a run in 14 of his 16 appearances and held batters to a .148 (8-for-54) average for Minnesota. ... The crowd of 31,562 pushed Cleveland's home ticket sales past 2 million for the 14th time in team history. The Indians didn't reach the milestone last season until Aug. 12.