Leo Carlsson and Mason McTavish scored in the shootout, and Lukas Dostal made 32 saves for the Ducks (33-23-3), who have won five in a row and are 12-2-0 in their past 14 games.
Joel Farabee and Yegor Sharangovich scored for the Flames (24-28-7), who have lost three of their past five games, including 2-0 at the Los Angeles Kings on Saturday. Devin Cooley made 34 saves.
The Flames took a 1-0 lead at 9:41 of the first when Farabee came off the bench with speed, split two Anaheim defenders and redirected a pass from Ryan Lomberg into the net.
Gauthier tied it 1-1 at 11:14 of the second period.
Cooley stopped two of Gauthier's shots before Gauthier banked the puck off Cooley's shoulder from below the goal line. The puck popped in the air, slid down Cooley's back and rolled into the net.
Sharangovich put Calgary up 2-1 with a power-play goal at 16:10, scoring bar down from inside the edge of the right face-off circle through traffic over Dostal's right shoulder.
Gauthier tied it 2-2 on the power play at 10:41 of the third period with a one-timer from the right circle that beat Cooley to the glove side off a touch pass from Jackson LaCombe.
