Michael Frolik and Stillman scored for the Panthers to tie the game in the third. It was Frolik's fourth goal in three games and Stillman's fifth point in his last four.
"We played a good game, and I thought we deserved two points," Panthers coach Peter DeBoer said. "It was a good team effort. We stuck to our system and stuck to our game plan, kept grinding away even though we weren't getting rewarded for it. We finally got a couple."
Minnesota had won the previous seven games over Florida, outscoring the Panthers 30-11.
Frolik knew that if they could sneak one shot past Backstrom, that streak might end.
"We said in the room, 'The first one is going to be the key. When we score the first one we can go forward and they can go back.' That's what they did," Frolik said.
Mikko Koivu and Robbie Earl scored first-period goals for the Wild, who found the back of the net twice on their first three shots.
The Wild were stuck on seven shots for about 12 minutes during the second and third periods. When Koivu managed to register the team's eighth shot, the crowd broke into a sarcastic cheer.
Despite Minnesota's lack of shots, coach Todd Richards felt his team played hard.
"I'll have to watch the tape, but my first impression is that Florida just did a good job," he said. "We were chasing them all night. Our best opportunities came in overtime."
Minnesota managed more shots on goal during the four-round shootout than it did in the second period.
Both teams entered the game eight points out of the final playoff spot in their respective conferences and long shots to reach the postseason.
"We've got to go out and win ourselves. It's in our control," said Stillman. "We can't count on anyone else to help us. This is a road trip where we need to get five out of six, if not six out of six points."
The Panthers' earned their third road victory in 11 games and first since Jan. 14. Florida will play its next two games at Colorado and San Jose.
Minnesota defensemen Brent Burns (hip) left the game in the third period. Shane Hnidy (leg) was forced out in the first. ... Wild RW Cal Clutterbuck missed the game because of a left leg injury sustained Sunday during Minnesota's loss to Calgary. Petr Kalus was called up from AHL Houston and made his Wild debut in Clutterbuck's place. ... Panthers D Keith Ballard, a Minnesota native, left Monday's practice due to a sore groin but played Tuesday. ... Columbus also totaled only 11 shots at home against Los Angeles on March 8.
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SAINT PAUL, MN - MARCH 9: Assistant coach Dave Barr of the Minnesota Wild signs a young fan's card prior to the...