"That's the way I score my goals," Holmstrom said. "Sometimes they go in, sometimes they go out.
Pavel Datsyuk added a power-play goal, and Darren Helm had an empty-netter for the Red Wings, who opened a five-game trip with their second straight win.
The Coyotes had a two-man advantage when Matthew Lombardi scored their lone goal.
Phoenix lost at home in regulation for the first time since Nov. 16, a span of 12 games, in front of a crowd announced as a sellout of 17,125.
"We know they've been playing real well here at home," Lidstrom said. "We wanted to play real solid defensively and try to wait for our chances. I thought we did an excellent job in our own zone."
The Coyotes lost their second in a row at home after their franchise-record 10-game winning streak there ended in a 3-2 shootout loss to San Jose on Thursday night.
"In the home building, we should have been able to pull it out," Phoenix forward Robert Lang said. "We started really good and obviously they kind of came on with the first goal and kind of took the wind away from us. But we came back and just made mistakes."
It was the first sellout since the home opener for the Coyotes, owned by the NHL after a bitter bankruptcy battle.
Jimmy Howard had 35 saves in his fourth straight start for the Red Wings, who have eight players out with injuries.
After failing to score more than three points in a game for 13 consecutive contests, Detroit has done it twice in a row.
The Red Wings scored first, seconds after their power-play try ended, on a penalty against Shane Doan for roughing.
Holmstrom, who had settled right in front of the net, deflected Lidstrom's straightaway shot past Bryzgalov with 5:16 left in the third period.
Phoenix tied it at 1 with a 5-on-3 advantage, Lombardi knocking it in on a nice pass across the crease from Lang 4:57 into the second period.
Detroit's Brett Ledba (slashing) and Helm (roughing) were in the penalty box when the Coyotes scored.
The Lidstrom-Holmstrom combination struck again with just 18.6 seconds left in the second period. Lidstrom's long shot from just in front of the blue line was deflected by Holmstrom in the crease again and it was 2-1 Detroit entering the third.
"He's just determined," Red Wings coach Mike Babcock said. "He's going to go there. It doesn't matter what they do to him, he's going to go back there. The thing about him that makes him good is he can tip the puck, he can score."
Coyotes coach Dave Tippett said the Red Wings still have a lot of talent, despite the injuries.
"The key pieces are still in place there," he said. "They got three top defensemen there. Datsyuk is as good a center man as there is in the world."
Doan was in the penalty box for interference when Datsyuk scored his 10th goal of the season on a left wrister with 14:09 left in the third period to make it 3-1. Todd Bertuzzi and Rafalski got the assists on the goal.
"I had a bad penalty," Doan said, "and you can't afford to take a penalty, no matter what the call is."
Phoenix's previous four games had ended in a 3-2 score -- the Coyotes winning two and losing two. ... Babcock said Howard would be back in goal Tuesday in Anaheim. ... Red Wings LW Brad May missed the game with a lower-body injury sustained in the team's home win over Colorado on Thursday night. ... Bryzgalov entered the game No. 2 in the NHL in goals allowed at 2.01 per game. ... Detroit leads the season series 2-1.