Darren Helm also scored, and Mrazek made 28 saves for the Red Wings (25-26-11), who have split the first two games of a five-game road trip that continues at the Edmonton Oilers on Saturday (10 p.m. ET; CBC, FS-D, NHL.TV).
"We want to win every game," Detroit forward Anthony Mantha said. "For sure, it gives us a big point in our chase to help the playoff run. We wanted that second point as soon as we started the overtime. We thought we had a good chance tonight. I think we played a pretty good game, a lot of shots on net. We need to build off of that for tomorrow night."
Helm put Detroit up 1-0 at 6:55 of the first period, but Versteeg tied the game at 8:19. Versteeg scored on a one-timer four seconds into a power play on a draw won by Sean Monahan. The assist gave Monahan 100 in his NHL career, and 200 points.
Tkachuk gave the Flames a 2-1 lead with an unassisted goal at 13:07.