Miettinen broke a scoreless tie with a power-play goal, taking a pass across the crease from Koivu and beating Mikka Kiprusoff on a tip-in with less than 5 minutes remaining in the frame.
The Wild quickly made it 2-0 with 53 seconds left in the middle period on Koivu's seventh of the year and Chuck Kobasew added an insurance tally in the third for Minnesota, which beat Calgary in the first of a home-and home series.
Niklas Backstrom turned aside 28-of-29 shots, including 14-of-15 in the third period to earn the win.
Matt Stajan got Calgary's only tally and Kiprusoff ended with 13 saves for the Flames, whose two-game winning streak was snapped.
2014: The Year in Music [PHOTOS]