Brossoit's shutout was his second of the series as the Oil Kings now have a 3-1 lead in their first-round matchup.
Curtis Lazar and Henrik Samuelsson had a goal and an assist each for Edmonton while Griffin Reinhart and Edgars Kulda had the others.
Mackenzie Skapski turned away 24-of-27 shots in defeat for the Ice, who have scored just three goals in the series.
Edmonton went 1 for 7 on the power play while Kootenay failed to score on five chances with the man advantage.
Game 5 is scheduled for Friday when the Oil Kings host the Ice at Rexall Place.