The Oil Kings took the best-of-seven series 4-1, outscoring the Ice 25-4 and blanking them on two occasions.
Curtis Lazar scored the game winner at 11:35 of the second period, streaking in on the left wing and putting a backhand shot over the blocker of Kootenay netminder Mackenzie Skapski.
Luke Bertolucci and Stephane Legault also scored for the Oil Kings while Lazar added an empty-net goal with less than two minutes remaining in regulation time.
"I thought we were a bit loose on the defensive zone coverage but at the end of the day I thought our guys did a really good job in the third period," said Edmonton's head coach Derek Laxdal. "It shows you how tough it is to finish that fourth game."
Jakub Prochazka had the lone goal for the Ice.
Laurent Brossoit turned aside 24 shots in net for Edmonton as Skapski stopped 32 for Kootenay.
Bertolucci scored the only goal of the first period after walking out from the corner at 13:07 and snapping the puck into the top corner.
Prochazka tied the game up at one apiece 2:51 into the second period, taking a feed off the wing from Jaedon Descheneau and tipping the puck in over the right pad of Brossoit.
The Ice came within inches of taking the lead on two separate plays with less than five minutes remaining in second, but couldn't take advantage of their opportunities. Jordyn Boyd snapped a shot on the odd-man rush just wide, then Sam Reinhart barely missed on a partial breakaway moments later.
"(It) doesn't matter what the series was at, we focused on one shift at a time," said Kootenay's head coach Ryan McGill. "The opportunity was there for us and we couldn't capitalize. You have to give credit to their goaltender and the experience on their team."
After Lazar put the Oil Kings ahead 2-1, Legault buried the insurance goal at 16:52 of the third period after tucking home a rebound on the backhand behind Skapski's outstretched left pad.
Edmonton failed to score on three chances with the power play while Kootenay went 0 for 4.
The Oil Kings move on to face the Medicine Hat Tigers in the second round. Edmonton took the regular-season series against the Tigers 5-1, with three contests decided by two goals or less and one needing overtime.
"A lot of credit to Kootenay, they had a great year and a great second half. We'll take a couple days and get ready for Medicine Hat," added Laxdal.