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.
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