Brossoit stopped 33 shots, including 18 in the second period alone, as Edmonton now leads the best-of-seven second-round matchup 2-0.
"A goalie doesn't mind getting a lot of shots in a row," said Brossoit. "It's when you get spurts and you sit there and get cold. Sometimes I'll be off my game and the boys will step up for me, so it's not frustrating at all, it's a team game."
The victory marked a personal nine-game winning streak against Medicine Hat for Brossoit.
Henrik Samuelsson and Cody Corbett had the goals for the Oil Kings, both coming on the power play.
"Our penalty kill was strong today, our power play was better," said Edmonton's head coach Derek Laxdal. "(Brossoit) was outstanding and he's probably been the difference in the first two games."
Cam Lanigan turned aside 32 shots in net for the Tigers.
Samuelsson opened up scoring on the power play at 6:04 of the first period when he deflected Martin Gernat's point shot in behind Lanigan.
Despite outshooting Edmonton 14-0 in the first 15 minutes of the second period, and 18-3 overall, the Tigers were unable to even up the score heading into the third.
"Today we played really hard," said Tigers' head coach Shaun Clouston. "Especially that second period we really got our feet moving. I thought our defense did a better job moving their feet and getting up ice.
"But when you get (seven) power plays, you'd like to get one. I thought Brossoit made a couple huge saves. We need to find a way to get one in the back of the net. We took a good step today, but not a big enough one and we have to take another."
Edmonton's power play struck again at 1:56 of the third as Corbett drove through the crease and whipped a spinning shot overtop a sprawling Lanigan to push the Oil Kings ahead 2-0.
Edmonton finished the night 2 for 4 with the man advantage while Medicine Hat failed to score on seven chances with the man advantage.
Medicine Hat will host Game 3 on Tuesday.