Anaheim trailed 2-1 in the final minutes of regulation when Andrew Cogliano equalised at 17:20 in the third to extend the battle and leave it to Fasth to carry them to victory.
The 30-year-old Swedish netminder made three saves during a scoreless overtime, finishing with 28, and denied Chicago's final two shootout attempts to improve to 6-0-0 in his magical first season.
Nick Bonino and Corey Perry converted shootout goals for Anaheim (9-2-1) as the Ducks won for the sixth time in seven games and moved two points clear of San Jose for first place in the Pacific Division.
Anaheim grabbed a 1-0 edge on Chicago on a Ryan Getzlaf wrist shot early in the second period, but the home Blackhawks quickly seized the lead with goals by Nick Leddy and Brandon Saad.
Despite the defeat, Chicago (10-0-3) are still without a loss in regulation and remain on top of the league's standings.
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