Trailing 3-games-to-1 and facing elimination, the Blackhawks erased a 1-0 deficit midway through the third period when Nick Leddy scored his first career playoff goal.
That forced a the fifth straight overtime in the Western Conference quarterfinal series, and Toews made sure it was a short one, firing a shot to the stick-side of Phoenix goalie Mike Smith that found the net less than 3 minutes into the tiebreaker.
Corey Crawford stopped 18-of-19 shots for the Blackhawks, who will again seek to avoid elimination when the series shifts back to Chicago for Monday's Game 6.
Gilbert Brule scored and Smith turned aside 36-of-38 shots for the Coyotes.
Millions of Getty images now available for free via embed tool
NBC reportedly holds celebs hostage to Jimmy Fallon's show