The lead didn't last long, though, as the Stars did what they've done best so far this season by scoring on the power play.
After a Filip Hronek slashing penalty 5:23 into the second period, the Stars wasted no time making the Wings pay, as John Klingberg scored four seconds into the power play on a slap shot from the point that deflected off Detroit defenseman Patrik Nemeth and past Greiss to tie the game, 1-1.
Joe Pavelski and Denis Gurianov earned assists on Klingberg's goal, giving Pavelski his 800th NHL point.
The power-play goal put the Stars at 9-for-14 on the man advantage to start the season, for a 64-percent clip. The Stars became the third team in NHL history to score nine power-play goals in their first three games.
Blashill said the Red Wings needed to be more disciplined against the Stars and limit the chances for their devastating power play.
"One of our major keys tonight was discipline," Blashill said. "Obviously, their power play is really humming. We've got to do a better job staying out of the box and play them five-on-five the best we can."
Much like the first period, the final frame on Tuesday night was a grind, with both teams struggling to maintain offense. The Red Wings earned a power play with 4:09 remaining in regulation when Anthony Mantha drew a hooking penalty, but despite several chances, Detroit came up empty, finishing regulation 0-for-3 on the man advantage.
The Wings battled early in the bonus period but Dickinson's goal at 1:32 of overtime ended the game and extended Detroit's road losing streak against Dallas to five games, with the Wings' last win in the Lone Star State coming in February 2016.
The Red Wings return to action against the Stars at the American Airlines Center Thursday night at 8:30 p.m. as Detroit wraps up its four-game road trip.