Jamie Benn's shot redirected off Collin Delia's right pad to Robertson, and the rookie forward scored on the rebound for the Stars.
"It's nice to finish the season off with a win, especially since overtimes weren't too great throughout the year," Robertson said. "Nice to overcome that and win the last game in OT. Weird feeling coming back to the locker room that you just won the game, but the season's over. It's mixed feelings right now."
John Klingberg had three assists, Robertson and Miro Heiskanen each had a goal and an assist, and Jake Oettinger made 19 saves for Dallas (23-19-14), which killed 3:40 of a double-minor penalty in overtime after losing 4-2 to Chicago here Sunday.
"For them to come out with that effort, it doesn't surprise me and it shows us what they're made of," Stars coach Rick Bowness said. "This team is exhausted, and they still found a way, and if we had to play tomorrow, they'd come up with the energy to come out and compete. Give them a ton of credit."
The Stars were 2-4-2 in their final eight games of the season.
Delia made 45 saves, and Alex DeBrincat, Vinnie Hinostroza, Dylan Strome and Pius Suter each had a goal and an assist for Chicago (24-25-7).
"Would've loved to get the win," Blackhawks coach Jeremy Colliton said. "Obviously, they were better than us tonight. We counted a lot on Delia. We got some opportunistic goals, allowed us to have the lead, but ultimately it just wasn't enough."
Chicago went 2-5-2 in its final nine games.
Dallas finished in fifth place in the eight-team Discover Central Division. Chicago was sixth.
Suter redirected Heiskanen's pass to DeBrincat, who scored on a breakaway with a backhand shot 53 seconds into the first period. DeBrincat had eight goals during a career-long seven-game goal streak, and 12 points (nine goals, three assists) during an eight-game point streak.
Suter received DeBrincat's pass at the right face-off circle and scored with a wrist shot to make it 2-0 at 1:22.
"I tried to forecheck and it goes off my stick, and I see [DeBrincat] going like the wrong way on the highway [to] just to get in the zone and he made a nice move," Suter said. "Next shift we get out there, basically make a nice cycle and I just put it on net and it went in."