Lidstrom beat Bryzgalov, who was screened by Holmstrom, with a slap shot with 7:28 left.
"I was aiming on the blocker side. There were two or three bodies in front of the net and I knew Holmer was one of them," Lidstrom said. "I just tried to pick the blocker side corner. If Holmer's not there, he's probably going to see the puck and make the save."
But Osgood still had to foil Doan, who got behind the Red Wings defense and was in alone from the blue line.
"I just tried to stay up as long as I could," Osgood said. "I tried to get everything low so he couldn't slide underneath me."
The Coyotes led 2-1 after two periods.
"We were a little bit rusty at the start. They had some jump on us," Lidstrom said. "I think it helped their cause by playing [Tuesday]. They got their legs going. It took us a while to get ours. Then we took the game over and created a lot of chances."
It was Detroit's first game after the All-Star Break.
"If you're them, you probably think you should have had a better fate," Detroit coach Mike Babcock said.
He was right.
"Detroit was lucky to get a point and lucky to get a win," Bryzgalov said. "We have to win this game."
It was the fourth time this season Phoenix failed to hold a 2-1 lead against the Red Wings.
"At 2-1, we had opportunities to get further ahead, but their goalie made some good saves," Coyotes coach Wayne Gretzky said. "It was disappointing not to get ahead 3-1, and it's frustrating that we had two giveaways in our end and they're up 3-2."
Holmstrom tied it 9:09 into the third period. Lidstrom's shot deflected in off of Holmstrom's knee for his 20th goal.
Jovanovski gave the Coyotes the lead 8:31 into the second period when he beat Osgood with a one-time slap shot from the left point for his seventh goal.
Zetterberg scored just 3:54 into the game after missing Detroit's previous two games and NHL All-Star festivities last weekend because of back spasms.
He missed the final 19 games of the regular season last season and five games in late December because of back problems.
Zetterberg's 29th goal came when his attempted pass from the left corner went in off of Bryzgalov.
Hanzal tied it for Phoenix at 9:57 when he beat Osgood from the slot for his sixth goal. It was the result of a giveaway by Red Wings defenseman Niklas Kronwall beside his own net.
Detroit defenseman Brett Lebda missed the game because of a left wrist injury. ... Gretzky's father, Walter, attended the game. ... Kronwall left the game with a left shoulder injury in the middle of the third period after being checked along the boards by Coyotes right winger Craig Weller. Babcock said the team wouldn't know anything definitive about the injury until Thursday.