Brenden Morrow had pulled the Stars even late in the third period with his second goal of the game.
Then, in the extra session, Eriksson beat Red Wings goaltender Chris Osgood on a three-on-one rush moments after the Stars failed to convert on a power play.
Eriksson added two assists for Dallas, which has collected at least one point in 12 of its last 14 games (10-2-2). Andrew Raycroft made 28 saves in the win.
Johan Franzen and Pavel Datsyuk both had a goal and set up another for the Red Wings, who have dropped three of four. Osgood stopped 28 shots and remains stuck on 399 career wins.
Trailing by a goal in the third period, Morrow, playing in his 700th career game, accepted a pass from Jamie Benn and slid a backhander between the pads of Osgood with 3:14 remaining.
Dallas battled back from a two-goal deficit in the first period.
Datsyuk converted an early power play with a backhander from the right circle, and Patrick Eaves stashed the rebound on a Niklas Kronwall shot to make it 2-0 at 12:25.
Brad Richards put the Stars on the board 1:31 later with a wrister that deflected past Osgood. Dallas tied the game at 17:57 as Morrow tipped a Trevor Daley slap shot over the left shoulder of Osgood for a power-play goal.
Franzen beat a screened Raycroft to restore Detroit's lead with 16.3 seconds left in the middle stanza.
Dallas improved to 14-2-3 in one-goal games this season...The Stars have scored a power-play goal in five straight games...Benn recorded two assists...Osgood is seeking to become just the 10th goaltender in NHL history to reach 400 wins and might get another chance to do so on in Detroit's next game, Wednesday against Vancouver.