Perry clanged the game-winner off the left post and Viktor Fasth turned away Brandon Sadd on Chicago's final attempt.
The victory was the second in a row for Anaheim. Chicago saw its winning streak end a four.
Fasth stopped 28-of-30 shots and won his sixth consecutive game. He is the first NHL goaltender to win his first six starts since Dan Ellis of Dallas in 2007.
Corey Crawford notched 29 saves for the Blackhawks. Sadd and Nick Leddy both scored in regulation.
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