Clutterbuck, who left the game in the third period with an upper-body injury, scored on a power play 7:16 into the frame to break a scoreless tie.
Eric Nystrom also found the net and Niklas Backstrom turned aside 37-of-38 shots for the Wild, who salvaged one win on a three-game southeastern road swing.
Dan Ellis let in three goals on six first-period shots and finished the game by stopping 14-of-17 Minnesota attempts for Tampa Bay.
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