Prucha's tally 8:16 into the third period gave the Coyotes a two-goal lead. After Johan Franzen responded with a goal midway through the period, Vrbata's snap shot past Jimmy Howard less than 2minutes later restored the cushion and iced the win.
Prucha and Vrbata also notched assists while Sami Lepisto and Wojtek Wolski got the other goals for Phoenix.
Ilya Bryzgalov stopped 29-of-31 shots for the Coyotes.
The Red Wings' only other goal came from Valtteri Filppula. Detroit had trouble penetrating the Phoenix defense to get shots on net, even though Phoenix was whistled for back-to-back penalties in the late going.