Brossoit earned his fourth shutout of the post season for Edmonton, which evened the best-of-seven Eastern Conference final 1-1.
Martin Gernat had two goals and three assists for the Oil Kings while Henrik Samuelsson scored once and set up two more.
Curtis Lazar, Ashton Sautner and Trevor Cheek each scored once for Edmonton.
"I don't think we're firing on all cylinders just yet," said Edmonton's Mitch Moroz. "I think we can be better five-on-five. Calgary did a good job of slowing us down in the neutral zone but we have a couple issues we need to address. We're definitely sharper (than game one), but we have some work to do before we head to Calgary."
Calgary's Jake Virtanen was handed a five-minute major and a game misconduct for kneeing Cody Corbett in the second period.
Chris Driedger turned aside 38-of-42 shots for the Hitmen while Mack Shields allowed two goals on eight shots in six minutes of relief.
Samuelsson opened up the scoring at 6:13 of the first, redirecting Gernat's saucer pass out of midair in behind Driedger.
Brady Brassart nearly tied things up for Calgary on a two-man advantage, deflecting a centering pass just wide of the post with five minutes left in the period, but Edmonton took a 1-0 lead into intermission.
Driedger came up big for the Hitmen in the opening minute of the second period, sprawling to make a left-pad save on T.J. Foster in tight.
Brossoit was forced to make a break-away save on Cody Sylvester at 7:38 as the winger broke in alone while killing off Virtanen's major penalty.
After being halted on nine power plays, Edmonton's man advantage clicked on its 10th attempt at 8:55 of the third as Gernat hammered the puck from the point past Driedger's blocker to make it 2-0.
Gernat then scored at 14:32, walking in and wristing it glove high on a two-man advantage.
Lazar knocked the puck out of midair 26 seconds later while still on the power play to make it 4-0, ending Driedger's night in goal.
Sautner and Cheek closed out scoring late in the third for the Oil Kings.
Edmonton finished the night 3 for 14 on the power play while Calgary was blanked on four opportunities with the man advantage.
"We obviously can't spend that much time in the penalty box," said Calgary head coach Mike Williamson. "It was a one-goal game... and our guys got unravelled. I thought at the end of the second we started to get some things going and start skating. Once they got their second goal, you could feel the frustration on the bench. Up to that point we were in the box too much but we did a good job killing them and earned some good chances to tie it up."
Game 3 is scheduled for Tuesday when the series shifts to Calgary's Scotiabank Saddledome.