Dylan Wruck had a goal and two assists for the Oil Kings (34-11-5), who extended their win streak to four games, and Michael St. Croix had a goal and an assist.
Stephane Legault scored once and Henrik Samuelsson chipped in with three assists to round out Edmonton's offence.
Jayce Hawryluk and Eric Roy scored for the Wheat Kings (18-28-4), who sit last in the Eastern Conference.
Tristan Jarry made 26 saves for the win as Curtis Honey turned away 47 shots in a losing cause.
Edmonton went 1 for 2 on the power play while Brandon failed to score on either of its chances with the man advantage.