The Wild won their fifth straight, a surge that has lifted them atop the Western Conference.
Minnesota trailed 1-0 heading into the final period, but they ran off three straight goals to take command.
Dany Heatley tied it 65 seconds into the period and Kyle Brodziak put the Wild in front at the 9:39 mark on the power play.
Clutterbuck scored what turned out to be the game-winning goal, producing a shorthanded tally with 6:41 left in regulation. Ryan Suter cut the Nashville deficit with 6:03 to play, but Josh Harding was perfect the rest of the way. He finished with 23 saves.
Regular Minnesota goaltender Niklas Backstrom missed the game for personal reasons, leaving the Wild having to scramble to ensure they had somebody to back up Harding.
The Wild brought up Matt Hackett from the AHL Houston Aeros, but there was some concern he might not arrive in time for the game, so Minnesota signed 51-year-old Paul Deutsch as an emergency goaltender.
Hackett, who has a 7-2-3 record with Houston, eventually reached the arena in time and Deutsch did not suit up for the game.
Pekka Rinne took the loss while making 27 saves.