The Coyotes (1-11-1) allowed two goals in the first 2:37, one night after ending an 11-game losing streak with a 4-3 overtime win against the Philadelphia Flyers.
"This was a huge lesson for us," Arizona coach Rick Tocchet said. "We dipped our toe in the water instead of diving in. We can't afford to ease into games."
WATCH: [All Coyotes vs. Red Wings highlights]
Jimmy Howard made 30 saves for the Red Wings (6-6-1), who led 4-1 in the second period and won their second straight game. Detroit ended a six-game losing streak with a 3-2 shootout win against the Florida Panthers on Saturday.
"This was a critical win," Red Wings coach Jeff Blashill said. "You don't want to call it a must win, but the schedule really sets up for you when you are at home and playing a team that got in late after playing last night."
Arizona pulled within 4-3 on Clayton Keller's goal with 3:17 left in the third period. Brandon Perlini waved at Keller's high shot and it hit off the shaft of Howard's stick, went over his shoulder and into the net. Perlini was originally credited with the goal, but Howard agreed with the revised call.
"I didn't think he touched it," Howard said. "To be honest, I thought it went in off the knob of my stick."
Darren Helm scored an empty-net goal with 1:22 left to make it 5-3.
Gustav Nyquist gave the Red Wings a 1-0 lead at 1:29 of the first period, intercepting Alex Goligoski's clearing attempt and beating Scott Wedgewood with a wrist shot. Nyquist's shootout goal on Saturday ended Detroit's losing streak.
"I wanted that one back," Wedgewood said. "I thought I had it lined up, even with the traffic, but it knuckled and dropped at the last second."