Mikko Koivu tallied with just one minute remaining in regulation for the Wild, sending the game into overtime.
Josh Harding let in a first-period goal, but kept Minnesota in the game with 20 saves over the next two periods and stopped 36-of-37 shots overall in the win.
The Red Wings hadn't lost five straight games since January of 2009. They have scored just five goals during their current losing streak, including none in Minnesota's 1-0 win in the opener of this home-and-home series Saturday.
Koivu tied it at 1-1 when he re-directed Justin Falk's wrist shot from the left point.
In the overtime period, Koivu maneuvered from the left circle to the high slot and made a nifty pass to Setoguchi, who positioned himself right of the crease.
Setoguchi's initial shot was stopped by Howard, but the puck caromed back to him and he lifted his next attempt past Howard's pad for the game-winner.
Minnesota finished 1-for-8 on the power play.
Earlier, Lidstrom re-directed Henrik Zetterberg's pass from the left point just underneath the crossbar to give Detroit a 1-0 lead on a power play, six minutes into the opening period.