The goals were the team-leading fourteenth and fifteenth for Holmstrom. Both came on net-front deflections, a Holmstrom specialty.
Matthew Lombardi evened things up at 5:53 of the second, taking advantage of a five-on-three for the Coyotes to get to the far side of Howard's net and bang in a shot off a cross-crease feed from former Red Wing Robert Lang.
Holmstrom put Detroit up again with 19 seconds left in the middle period, deflecting another Lidstrom shot.
Pavel Datsyuk beat Bryzgalov with a shot from the slot into the top corner of the net on a Red Wings power play 5:51 into the third period and Darren Helm added an empty net goal to wrap up the scoring.
Detroit finished the night one-for-four on the power play, as did Phoenix.
Bryzgalov stopped 32 of 35 shots against.
The Red Wings continue their season-high five-game road trip on Tuesday in Anaheim.
Notes: The game was the 900th of Todd Bertuzzi's career.
Credit: Clark Rasmussen