Christensen opened the shootout with a deke to the right before moving back to the left and slipping the puck past Jose Theodore, and after Florida's Kris Versteeg failed to convert, Matt Cullen ripped a shot through Theodore's pads from the slot.
Cullen also scored in regulation, while Kyle Brodziak tallied Minnesota's other goal. Backstrom stopped 41-of-43 shots to earn the win.
Bergenheim and Mikael Sameulsson lit the lamp for the Panthers, who took a one-point lead over the Winnipeg Jets for the top spot in the Southeast Division with the point. The Jets are also in action Thursday as they are hosting the Tampa Bay Lightning.
Theodore made 26 saves as Florida lost four consecutive games for the first time since recording an 0-2-2 mark from January 13-20.
The game entered the third period knotted at 1-1, and the Wild used an early power play to take the lead.
Stephen Weiss was called for tripping just 42 seconds into the third and Minnesota took advantage 32 seconds later when Dany Heatley's one-timer from the right circle was blocked, but Cullen grabbed the rebound in the slot and flicked it into the net.
Florida, though, was able to force the extra period when Bergenheim grabbed a pass to the right of the net and fended off a defender before spinning around and sweeping a shot into the far side of the net with 4:12 left in regulation.
Backstrom made four saves and Theodore stopped two shots in overtime.
The Panthers broke a scoreless tie thanks to a Devin Setoguchi goaltender interference penalty in the second frame.
Florida lost the ensuing offensive zone faceoff, but after Minnesota failed to clear the puck, Weiss' shot from the right circle hit the post and bounced right to Samuelsson in the left circle, where he fired it into the open net with 6:37 left in the period.
Minnesota, though, sent the game into the second intermission tied at one on Brodziak's tally with 20 seconds left in the middle stanza.
Heatley attempted a one-timer from the high slot, but his stick shattered on the attempt. Marek Zidlicky redirected the slow-moving puck on net and Theodore kicked it aside, but Brodziak followed the play and cleaned up the rebound.
Weiss was playing his 614th career game with Florida, setting a new franchise record...Brodziak's goal was his career-high tying 16th of the season...Minnesota improved to 13-14-5 on the road this season, while Florida fell to 14-8-8 at home...Panthers defenseman Ed Jovanovski returned from a 14- game absence with a broken right hand. He had a plus-1 rating and two penalty minutes in 18:22 of ice time.