Moore was stopped by Frederik Andersen on a breakaway in overtime but got the winner in the sixth round of the shootout on his 21st birthday.
Spencer Knight made 28 saves, and Ilya Mikheyev had a goal and an assist for the Blackhawks (21-22-7), who have won two in a row.
Joel Nystrom, Jordan Staal and Jackson Blake scored for the Hurricanes (31-15-5), who are 3-0-1 in the past four. Andersen made 19 saves.
Mikheyev scored short-handed to give Chicago a 1-0 lead at 11:12 of the first period. He kept the puck on a 2-on-1 with Ryan Greene and was stopped by Andersen, but the Chicago forward tapped in the rebound on the backhand.
Carolina tied it 1-1 at 12:50. Nystrom took a backhand pass from Jesperi Kotkaniemi and scored on a short-side snap shop shot from the top of the left circle for his first NHL goal in his 36th game.
The Blackhawks went ahead 2-1 at 4:35 of the second period. Ryan Donato took a no-look backhand pass from Moore and made a quick feed to Nick Lardis, who one-timed a shot from below the left hash marks.
Staal made it 2-2 at 9:16. The Carolina captain forced a turnover against Artom Levshunov in Carolina’s defensive zone and skated the puck through the neutral zone before scoring on a snap from the right circle.
Chicago took a 3-2 lead at 13:38 when Conner Murphy took a cross-ice pass, stick-handled to the top of the crease and wrapped the puck around Andersen’s left pad.
The Hurricanes tied it 3-3 at 14:20. Logan Stankoven circled the net and passed across the crease for a back-door tap-in by Blake at the right post.
