Cogliano gave the Oilers their first lead at 2-1 when he tipped in Brule's shot early in the third period. Then, with less than 8 minutes left to play, Brule sailed a wrist shot off a feed from Cogliano past Brian Elliott and Edmonton went on to beat Ottawa for the first time in the last seven meetings between the teams.
Tom Gilbert scored in the first period and Taylor Hall added an empty-netter for the victors.
Martin Gerber earned a win in his first start of the season after being recalled from Oklahoma City, turning away 22-of-23 shots.
Nick Foligno got Ottawa's only goal and Elliott rejected 21-of-24 Oilers shots.