Minnesota had established an early 2-0 lead when Justin Morneau and Trevor Plouffe got aboard with two outs in the third inning and Hicks followed by clouting a pitch into the left-field stands, breaking the contest open.
Pedro Florimon added a solo shot and Morneau went 3-for-5 with an RBI for Twins, who after dropping six straight to the Yankees dating back to last season took a second consecutive win from New York, claiming the three-game series.
Minnesota took advantage of a pair of Yankees errors to score five unearned runs en route to the triumph.
Rookie Kyle Gibson (2-2) collected his second major-league victory, allowing three runs (two earned) on five hits over five innings. He overcame four walks and struck out one.
New York's CC Sabathia (9-8) was tagged for eight runs (three earned) on eight hits, lasting just four full frames.
Ichiro Suzuki went 3-for-5 with a solo homer and two RBI for the Yanks.