Robertson scored twice, and Roope Hintz had two assists for the Stars (42-27-5), who are 3-1-2 in their past six. Wedgewood made 24 saves.
"We'll take a point where we can get it," Robertson said. "This time of the year, you'd rather one point than zero points, but we still have to keep going and we have an important one Saturday (against the San Jose Sharks)."
The Stars moved into a tie for the first wild card from the Western Conference with the Nashville Predators, who lost 4-0 to the Edmonton Oilers on Thursday.
"We outplayed them all three periods, we just didn't put enough pucks behind them," Dallas coach Rick Bowness said. "We had goal posts, we had open nets, we didn't get the puck in. That's hockey. I'm very happy with the way we played. We just beat Tampa (1-0 on Tuesday), we just tied Minnesota and we lost in overtime, two of the best teams in the league. We keep playing like that, things are going to be OK."
Fiala gave Minnesota a 1-0 lead 2:14 into the first period when he received a centering pass from Gaudreau, settled the puck onto his backhand, and scored past Wedgewood as the goalie was moving in the opposite direction.
Robertson tied it 1-1 at 6:02 of the second, scoring off his own rebound from the edge of the crease on the power play.
Fiala scored shorthanded at 11:36 to put the Wild back in front 2-1. After Alexander Radulov turned the puck over at his own blue line, Fiala took another pass from Gaudreau, split the defense and scored on a breakaway.
It was his first shorthanded goal in the NHL.
"It was my goal to be a complete player," Fiala said. "Now playing PK, very grateful to get the chance and I feel very comfortable in that position. Just great to do something else for the team, to be defensively and kill penalties is very big."