Hernandez (3-1), recalled from Class AAA Rochester before the game, overcame six walks and three hits to yield just a pair of runs, striking out two.
Josh Willingham supplied a pair of run-scoring hits and while Oswaldo Arcia and Trevor Plouffe delivered back-to-back RBI singles in the fifth, putting the Twins ahead to stay.
Tribe starter Carlos Carrasco (0-3) was touched for three runs on six hits over 4 2/3 frames. He issued four walks in the loss.
Drew Stubbs homered and Jason Kipnis went 3-for-4 with an RBI for the Indians.
McPhee, Cokas 'working on their marriage' after affair
Beautician charged with giving client fatal silicone butt injection