Brad Marchand scored 35 seconds into overtime, giving the Boston Bruins a 3-2 victory.
It was the Red Wings' sixth consecutive loss in overtime this season.
Boston tied it 2-2 on David Pastrnak's 15th goal at 18:34. The Bruins pulled goalie Tuukka Rask, and with the extra skater, Brad Marchand found Pastrnak alone at the post for an easy tap past goalie Jimmy Howard.
The Red Wings had taken a 2-1 lead on Dylan Larkin's shorthanded goal at 11:34.
Darren Helm gathered a loose puck and fed Larkin streaking alone down the middle of the ice.
Larkin froze goalie Tuukka Rask with a move, and neatly deposited his fifth goal, giving the Red Wings a 2-1 lead.
It was the Red Wings fifth shorthanded goal of the season — two more than they had all last season.
The Bruins (15-9-4) appeared to be gaining momentum, which was bad news for a Red Wings team that had won only once in the last 10 games.
And, sure enough, ultimately it was another discouraging loss for the Red Wings (11-13-7), who sink deeper in the Atlantic Divisions standings.
Noel Acciari tapped a loose puck past goaltender Jimmy Howard at 3:02, his second goal, after the Red Wings coughed up control of the puck after winning the faceoff.
Tatar scored at 9:15 of the second period, breaking a scoreless tie.
The Red Wings had been 0-for-13 over their last four games on the power play, and a dreadful 2-for-22 over seven games.
Tatar got the puck near the dot, and snapped his ninth goal of the season, with Justin Abdelkader providing the screen in front of goalie Tuukka Rask.
Just after Tatar's goal, Howard denied David Pastrnak on a breakaway after Patrice Bergeron found Pastrnak past the Red Wings defense.
The Red Wings outshot Boston 9-2 in the first period, not allowing a Bruins shot over the last 18:17.
It was a season-low allowed for the Red Wings, who allowed three shots Dec. 5 to Winnipeg.