Robidas' shot from the center of the blue line got past screened goalie Jose Theodore while Minnesota's Brent Burns served a tripping penalty.
Adam Burish put Dallas ahead early in the third period, Karlis Skrastins notched his first goal of the season, and Steve Ott scored during a power play for the Stars, 8-0-1 in their last nine at home.
Kari Lehtonen made 28 saves for Dallas.
Antti Miettinen, Chuck Kobasew and Martin Havlat scored for Minnesota, which rallied from deficits of 2-0 and 3-2.
With the score tied at 2, Burish set up in front of the net and backhanded Trevor Daley's centering pass by Theodore at 5:57.
But Havlat tied it at 3 with 3:09 left in regulation from in front.
Minnesota is 1-5-2 in its last eight games and 0-9-3 in its last 12 trips to Dallas. The Wild haven't won in Texas since March 21, 2003.
Skrastins gave the Stars momentum 3:42 in when his shot from just inside the blue line eluded a screened Theodore. Skrastins notched his 30th career goal in his 783rd NHL game, 7 seconds after a Dallas power play expired.
Lehtonen preserved the one-goal lead when he denied Cal Clutterbuck on a short-handed breakaway less than 2 minutes into the second period.
Theodore made a big save, too, stopping James Neal on a solo rush off a power play at 11:48 of the second period.
But Ott pushed the advantage to 2-0 at 11:45 of the second with a wrist shot from the right circle that was set up by Neal.
Former Stars forward Miettinen made it a one-goal game at 17:34 of the second period off a scramble in front. Kobasew's rebound goal tied it at 2 at 1:31 of the third.
NOTES: Kobasew, activated from the injured list on Friday, had missed 14 games because of a groin injury. ... Dallas recalled D Philip Larsen from AHL Texas and placed RW Krys Barch (foot) on the injured list. ... Lehtonen earned his second assist of the season when Ott scored. ... Theodore has allowed at least three goals in 12 of his 16 outings against the Stars.