The Indians trailed 6-0 after five innings but scored five times in the sixth when five consecutive Cleveland batters delivered run-scoring hits.
Matt Capps came in to pitch and Duncan greeted him by pulling a 3-2 offering into the left field corner for the go-ahead runs.
Nick Hagadone (1-0) collected his first major league victory with a scoreless seventh and Chris Perez locked it down with his 36th save for the Tribe.
Ben Revere went 3-for-5 with an RBI for the free-falling Twins, who lost for the 14th time in 15 games.
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