The Blackhawks won their fourth in row and have managed at least one point in 17 of their 19 games this year.
Chicago let an early two-goal lead get away and found itself trailing heading into the third period. Pirri, however, deadlocked the game just three minutes into the stanza with a deflection.
He scored again in the fourth round of the shootout to end the Coyotes' brief winning streak at two games.
Michael Stone had two goals for Phoenix.
Aaron Carter is still in love with Hilary Duff
Ray Liotta sues skin care company over use of likeness