McAllister (7-7) corralled his third win in three starts, stopping the Twins on a pair of runs on four hits. The right-hander issued two walks and struck out five.
Carlos Santana put the Tribe ahead for good with a two-run homer in the first inning and Jason Kipnis (2-for-3, two runs scored) added a two-run shot in the third for a 4-1 lead.
With their fourth win in five contests, the Indians remained three games behind American League wild-card leader Oakland.
Twins starter Liam Hendricks (0-2) was pounded for seven runs on seven hits over 4 1/3 innings.
Brian Dozier cracked a solo homer and drove in both Minnesota runs.
Notable deaths of 2014 [PHOTOS]