Connor Zary scored for the Flames (27-34-7), who ended their two-game losing streak.
Joel Farabee and Matvei Gridin scored in the shootout and Cooley stopped Jake Neighbours and Jimmy Snuggerud to clinch the win.
Dylan Holloway scored for the Blues (27-30-11), who have lost two straight and three of five (2-1-2). They are six points back of the second wild card into the Stanley Cup Playoffs from the Western Conference. Joel Hofer made 30 saves.
Zary gave the Flames a 1-0 lead at 2:34 of the first period, taking a pass from Ryan Strome and backhanding the puck past Hofer's glove as he skated into the slot. It was the first goal and point in 11 games for Zary.
Martin Pospisil appeared make it 2-0 at 12:43, but St. Louis challenged successfully challenged for offside after video review determined Calgary defenseman Kevin Bahl did not hold the puck in the offensive zone.
The Blues then challenged a goal by Yegor Sharangovich at 17:23 for a missed stoppage in play. Video review determined that Zary had played the puck with a high stick, negating the goal.
Holloway tied it 1-1 at 3:47 of the second period when he finished Snuggerud's pass on a 2-on-1 with a snap shot from the left face-off circle that squeaked under Cooley's left arm.
Farabee thought he put Calgary up 2-1 with a snap shot from the right circle at 9:38, but the call was reversed following a successful challenge for offside.
The Flames outshot the Blues 7-3 in overtime, but were unable to capitalize on a power-play opportunity with 2:37 remaining in the extra frame after Pavel Buchnevich was called for high-sticking Zary.
