Backstrom made brilliant stops on Anze Kopitar and Jack Johnson in the tiebreaker before Bouchard scored on Bernier after nearly losing control of the puck on the way in to the net.
Backstrom then completed the victory, and his third shutout of the season, by stopping Dustin Brown and the Wild posted their fifth win in six games to move ahead of the Kings into the eighth and final playoff spot in the Western Conference.
Backstrom stopped all 27 shots for Minnesota.
Bernier posted his first shutout of the year and the second of his career, turning aside 25 shots for the Kings.
Notable deaths of 2014 [PHOTOS]