Alex Goligoski scored to make it 3-1 at 7:06 when his shot from along the left boards went under Forsberg's pad. Forsberg, who made 10 saves, was pulled following the goal and replaced by Jeff Glass, who made six saves.
"The third goal, I think that was the one that you need a save made," Blackhawks coach Joel Quenneville said. "We had some momentum there in the game. That one has to be stopped."
Tobias Rieder extended the lead to 4-1 at 9:33 when he scored on a rebound in the crease after Glass couldn't locate the puck following Christian Fischer's shot.
Nick Cousins made it 5-1 at 7:12 of the third period, redirecting Goligoski's shot from the point.
Christian Dvorak scored on a breakaway at 15:38 to make it 6-1.
"I just like to see some guys scoring," Coyotes coach Rick Tocchet said. "Hopefully it gets them some confidence and loosens some people up."
This is the latest the Blackhawks have been .500 since Feb. 13, 2008 (25-25-6).
"I don't think we want to be fooled knowing that you out-attempt them, you outshoot them, and out-chance them, and you lose the game without even giving yourself a chance of getting two points," Quenneville said. "That's what we play for. We're in the winning business."