With the game tied 2-2 early in the final frame, the Coyotes were whistled for having too many men on the ice and the Red Wings made them pay.
Henrik Zetterberg's pass through the crease hit the skate of defenseman David Schlemko and bounced straight to Datsyuk, who fired it into an empty net for the decisive tally.
Jonas Gustavsson turned aside 29 shots, including a third-period penalty shot by Lauri Korpikoski.
Martin Hanzal and Rob Klinkhammer found the net for the Coyotes, who dropped a third straight.
Jason LaBarbara made 32 saves for Phoenix.