The Red Wings savored the victory in a building they used to visit far more often.
"I was saying that the other day," Detroit goalie Jimmy Howard said. "I miss playing these guys so much. It's always a spirited contest. How many times have we gone to shootouts, you know? It seems like almost every single game it comes right down to the skills competition. It's always fun when you get to play these guys."
"It's not ideal giving a team like that, with so much skill and so much talent, chance after chance, and trying to stay in the hockey game," Howard said. "We found a way to take a lead, gave it up real quick, found a way to get the lead late in the game, gave it up real quick [again], but the only thing that matters is we're walking out of here with two points."
Detroit (32-14-10) ended a three-game losing streak starting a six-game road trip against the Western Conference. Tatar and center Darren Helm scored regulation goals for the Red Wings.
"It's important," Detroit coach Mike Babcock said. "You don't want to keep losing. You want to get back on track. We were fortunate. We had some guys really dig in for us, including [Howard], but we've got to get better than we were tonight."
Chicago (35-18-5) earned a point for the seventh straight game but is 2-0-3 in the first five of their eight-game homestand.
"I thought we kept a really good team's chances to a minimum," Versteeg said. "There's a lot of building blocks to build upon here. Just keep going forward. There's a lot of games coming up and [we'll] try to string together a few wins again."
Jonathan Toews scored the shootout goal for Chicago.
Tatar and Saad scored 39 seconds apart near the end of the second period. Tatar scored his 24th goal off a backhand feed from Helm at 16:10, and Saad scored his 18th off a wrist shot at 16:49 to extend his point streak to four games.
The two third-period goals came in the final 2:06. Helm broke a 1-1 tie at 17:54 before Versteeg tied it 2-2 at 18:37 with a goal credited to him. His shot from the left circle appeared to hit Howard's glove, then Toews crashing the net before hitting Detroit defenseman Niklas Kronwall and bouncing into the net.
"The first period we scrambled a little bit, but I thought we got better in the second and third," Kronwall said. "[Howard] was huge for us, and then I score that goal to make it interesting [at the end]. Of course, it feels a lot better and it's a lot easier to take when [Howard] comes up big in the shootout."
Detroit swept the season series after winning 4-1 at Joe Louis Arena on Nov. 14.