Nicklas Lidstrom and Dan Cleary each had a goal for the Red Wings, who have won four of their last five games and pulled within a point of the first-place St. Louis Blues in the Central Division. Jimmy Howard stopped 29 shots in the win.
It was the Red Wings' first game at home after a three-game road trip. They traveled all day Sunday after dropping a 6-4 decision in Vancouver on Saturday night.
"We know this game is a tough one, both in the legs and mentally," Zetterberg said. "We knew it was not going to be a beautiful win if we were going to get the W. It was a pretty ugly one. We're taking it and happy to get the day off tomorrow."
Martin Hanzal and Keith Yandle each had a goal for the Coyotes, who have lost three straight. Bryzgalov made 37 saves for Phoenix, which fell to 0-5 in overtime this season.
"We worked hard, had our chances. Both goalies played well," Phoenix defenseman Ed Jovanovski said. "Sometimes those are the toughest."
In overtime, Todd Bertuzzi took a pass from Zetterberg just after Bertuzzi raced over the blue line along the right wing boards. Bertuzzi then made a cross-ice pass to Ruslan Salei, who moved in to the top of the left circle and aimed a shot at Zetterberg's stick. The puck hit the target and trickled through Bryzgalov's pads and into the net.
It was Zetterberg's fourth goal.
"Bert made a good pass to Rusty and he really held onto the puck, so I had a chance to get to the side of the net," Zetterberg said. "He made a good shot pass, I just put the stick there and it went in."
Yandle gave the Coyotes a 2-1 lead 7:31 into the second period when he picked up a loose puck in the lower left circle off the rush and beat Howard. It was Yandle's third goal of the season.
Lidstrom's second goal of the season tied the game at 7:24 of the third period on a slap shot from the left point.
"Down that goal going into the third, it was good to battle back," Detroit defenseman Brian Rafalski said.
Cleary opened the scoring 1:40 into the game when he put in the carom off the end boards of Rafalski's wide shot into an empty net. It was Cleary's fourth goal and third in three games.
Hanzal took advantage of a favorable bounce at the other end to tie the game with 55 seconds left in the opening period. Derek Morris' shot from the point went wide but came off the end boards between Howard and the goal post and Hanzal poked it into the net. It was Hanzal's second goal, both against Detroit.
"We have a lot of guys playing very hard," Phoenix coach Dave Tippett said. "But we need to make that extra play or take that one extra mistake out of our game that's been critical to us. Right now, it's that one situation that's taking the game away from us. We're making some mistakes to beat ourselves."
Material from wire services and team media was used in this report.
1 - 0 DET
1 - 1 Tie
2 - 1 PHX
2 - 2 Tie
3 - 2 DET
Interference on goalkeeper