Anaheim trailed 2-1 in the final minutes of regulation when Andrew Cogliano equalized 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.
"You always work hard to get to play and when you play you work hard to try to get the wins," Fasth said. "I don't take anything for granted and I've got to work hard every day to get my chances.”
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.
The San Jose Sharks fell to Nashville Predators 1-0 in overtime on Tuesday to lose ground.
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.
Cogliano's tying goal came as he tipped a shot from Ben Lovejoy.
"I was at the right place at the right time," Cogliano said. "I think it was just fitting for the kind of night we had.”
Blackhawks goaltender Corey Crawford made 29 saves to keep pace with Fasth who was just a little better in the extra sessions.
Despite the defeat, Chicago (10-0-3) are still without a loss in regulation and remain on top of the league's standings.
The Blackhawks were returning home following a six-game road trip while the Ducks are in the middle of a similar road stint.