The Stars jumped out to a 2-0 lead early in the first period and were up 3-1 at one point before the Flames stormed back with three straight in the second period.
Dallas would tie it up before Rasmus Andersson put them ahead in the third period, only for the visitors to again knot the game and force OT.
Elias Lindholm, MacKenzie Weegar, Blake Coleman and Nick Ritchie also scored for Calgary, while Andersson finished the night with three points, as did fellow blueliner Weegar.
Jacob Markstrom got the start, turning aside 20 shots while at the other end of the ice, Jake Oettinger made 33 saves.
It was lively OT frame, with Andersson hitting a post and Markstrom denying Stars captain Jamie Benn when he held the puck on a 2-on-1 before Robertson ended it.
The Stars opened the scoring 1:05 into the tilt, Roope Hintz getting a shot on Markstrom, the Flames 'tender making the initial save but there was a pileup as he lay prone, the puck eventually making it up and over him, Joe Pavelski credited with the goal.
Dallas made it 2-0 just 52 seconds later, Hintz the man to get the first shot as he stood all alone out front, again Markstrom making the stop but Radek Faksa getting the rebound and putting it home.
But Lindholm would answer back 63 seconds ticks after.
Tyler Toffoli got the puck to Andrew Mangiapane who walked down main street and fired a shot, Oettinger kicking the rebound out right to Lindholm who one-timed home his 20th of the campaign from a tight angle.