Brassart took the puck off the stick of defender David Musil in front and whipped the puck into the wide open net 1:44 into overtime.
Victor Rask scored for Calgary in the first period, and Zane Jones made it 2-0 in the second.
Trevor Cheek got one back for Edmonton in the second period, and Henrik Samuelsson tied it for the Oil Kings in third.
"Right from the start you could tell it was going to be pretty intense," said Hitmen coach Mike Williamson. "I think our guys are experienced enough now in close games that (despite giving up the late goal) we know to go in the dressing room to reset and refocus, and that anything can happen."
Rask opened the scoring 1:25 into the game, streaking in on the left wing and floating a wrist shot over the left pad of Edmonton netminder Laurent Brossoit.
Edmonton came within inches of tying the game at 13:43 of the period as Cheek took a centring pass from Michael St. Croix, looking to pick a shot just under the crossbar. But Calgary netminder Chris Driedger fought the shot off with his shoulder.
Calgary made it 2-0 on the power play at 3:32 of the second period, needing just seven seconds to capitalize off a faceoff win as Jones deflected Spencer Humpries' point shot in behind Brossoit.
Driedger came up big in net again for Calgary on the penalty kill, turning aside Cheek in tight on a rebound with a sprawling right pad save halfway through the frame.
But third time was the charm for Cheek, who was left alone in the slot and buried a wrist shot into the high glove corner at 13:42 of the middle stanza to pull Edmonton back within one.
Edmonton's power play struck at 13:05 of the third with Samuelsson finishing off a crisp tic-tac-toe play at the side of the net, leaving Driedger no chance as the Oil Kings tied the game 2-2.
Having lost the home-ice advantage, Edmonton coach Derek Laxdal was focused on the necessity to come back strong in game two on Friday.
"We faced a bit of adversity with Kootenay and obviously guys don't feel good about the result tonight," said Laxdal. "Calgary isn't going to lie down, they're going to come in here and play harder. They have a chance to come in and take two games out of here. We have to be better in on the forecheck, I don't think we got first to the puck enough. I'm not going to name names, but some guys have to be sharper and step up (Friday)."
Calgary was 1-for-4 on the power play while Edmonton went 1-for-5 with the man advantage.
Chris Driedger stopped 33 shots for the Hitmen, while Brossoit turned aside 16 for the Oil Kings.