Pospisil scored from the left face-off circle off a rush to give Calgary a 3-2 lead.
Yegor Sharangovich then scored into an empty net with 1:22 left for the 4-2 final.
Aydar Suniev had a goal and an assist, and Adam Klapka also scored for the Flames (1-5-0), who ended the longest losing streak to start a season in franchise history. Dustin Wolf made 23 saves.
Beckett Sennecke and Leo Carlsson scored, and Lukas Dostal made 32 saves for the Ducks (4-2-0), who were coming off a 4-2 win in Calgary on Saturday, their second straight victory.
Anaheim forward Roger McQueen, a first-round pick (No. 10) in the 2025 NHL Draft, made his NHL debut. He had one shot on goal in 11:26 of ice time.
The Ducks surrendered the first goal for the first time this season when Klapka scored at 8:32 of the first period to give Calgary a 1-0 lead.
Anaheim forward Jeff Malott failed to connect with Tim Washe on an outlet pass and the puck went to Maxim Tsyplakov in the slot. He dropped the puck for Klapka, who scored with a snap shot from above the hash marks.
Sennecke tied it 1-1 at 12:09 after he stole the puck from Calgary defenseman Simon Nemec at his own blue line during 4-on-4 play and scored on a breakaway.
Carlsson gave the Ducks a 2-1 lead on the power play at 14:05, scoring with a snap shot from near the right face-off dot shortly after an 11-second two-man advantage came to an end.
Suniev tied it 2-2 at 8:02 of the second period on a one-timer from the inside edge of the right circle, eight seconds after a two-man advantage that began at 5:54 expired.
Suniev scored his first NHL goal in the loss to the Ducks on Saturday.
