Albers (2-0), who began his big-league career last week with 8 1/3 scoreless innings against the Kansas City Royals, allowed only a fourth-inning single to Jason Kipnis and an eighth-inning single to Yan Gomes en route to the victory.
The 27-year-old Canadian left-hander didn't surrender a walk and struck out two, setting a Twins club record for most scoreless innings to begin a career.
Trevor Plouffe socked a two-run homer and Brian Dozier contributed a solo shot for Minnesota, which took the opener of a three-game series.
Cleveland's Danny Salazar (1-1), making just his third major-league start, was touched for three runs on three hits over four innings.
The reeling Indians dropped their seventh in eight games.