Niklas Hagman scored the game-winner in the 2 decision.
"We really wanted these two points. It's really important for our team and I really wanted it," said Hiller.
Miikka Kiprusoff was denied his 300th career win despite making 31 saves.
Calgary trailed 2-1 entering the third period, and Anaheim's Lucas Sbisa was handed a five minute major penalty and a game misconduct for head butting 2:40 into the frame after his head collided with the head of Calgary's Tim Jackman on an open-ice check. The Flames took advantage with the game-tying goal
Iginla split Anaheim's defenseman while receiving a pass at the Ducks' blue line and raced in on a short breakaway, skating into the slot and sending a wrister past the stick side to knot the game at two at the 6:15 mark.
Each team was then awarded a late power play in the third period, but neither converted.
Early in the extra period, Anaheim's Teemu Selanne just missed the game-winner as he hit a post from the right circle after Kiprusoff lost track of the puck.
Iginla and Tanguay also failed to end the game when they came down the ice on a 2-on-1 rush, but Iginla missed the net as he lifted the puck too high over the crossbar.
"The game was more of a roller coaster ride. We battled back to get a point," Flames head coach Brent Sutter said. "In the shootout, you have good goalies making great saves and good players making great moves. They were fortunate to get one more past our goalie."
Ryan broke a scoreless tie with 4:08 remaining in the first period when he settled a rolling pass in the slot before snapping a shot through a screen and past Kiprusoff.
Then nearly two minutes later, Sheldon Brookbank ripped a one-timer from the right point that was redirected into the top-left corner by Belesky to put the Ducks ahead 2-0 after one.
But the Flames cut the deficit to one early in the second period. Blair Jones received a pass to the right of the net before sweeping behind the goal and sending the puck out in front from the left post. Tanguay grabbed the pass in the slot and slid a shot through the legs of Hiller at the 4:41 mark.
Anaheim then had several chances to extend its lead in the second as the Ducks were awarded four power plays, including a two-man advantage in which they hit two posts, but they could not convert and held a 2-1 advantage into the final period.
Five of the last six games between the teams in Orange County have been decided after regulation...Selanne currently sits at 655 career goals, which is one goal behind Brendan Shanahan for 12th on NHL's all-time list...Calgary failed to record a shot in overtime...The Ducks went 0-for-5 on the power play, while Calgary was 1-for-4 on the man advantage.