Barrie scored both of his points on the power play off of his slap shot from the point.
Jesse Puljujarvi and Alex Chiasson also scored for Edmonton, who moved to 26-15-2.
Connor McDavid and Leon Draisaitl each had two assists.
Mike Smith earned the shutout turning aside 26 shots for the Oilers.
Connor Hellebuyck made 22 saves for the Jets, who dropped to 27-15-3.
There was no scoring in the first period.
Tyson Barrie opened the scoring 12 minutes into the second on the power play when he beat Winnipeg goaltender Connor Hellebucyk between the pads with a slap shot from the point.
Barrie hit Connor McDavid with a pass along the boards which allowed him time to move into shooting position before McDavid returned the puck to him, which Barrie one timed past Hellebuyck.
The goal was his 6th of the season.
Jesse Puljujarvi made it 2-0 6:37 into the third when Connor McDavid found him alone in front of the Winnipeg net.
Jets winger Kyle Connor left Puljujarvi alone in front to cut of McDavid along the end boards.
McDavid then hit Puljujarvi with a no-look backhand pass which Puljujarvi promptly hammered past Hellebuyck to score his 10th of the season.
With Nikolaj Ehlers off for cross checking, Alex Chiasson tipped a Tyson Barrie shot from the point past Hellebuyck to make it 3-0.
The goal was his 7th of the season.
Winnipeg pulled their goaltender late but were unable to score.
The Oilers return home to host Montreal on Monday while Winnipeg hosts Toronto Thursday.
SHOTS-EDMONTON 25 WINNIPEG 26
NOTES-Jets Captain Blake Wheeler was cleared to play earlier in the day but was held out of the lineup in order for him to get some practice before facing the Maple Leafs on Thursday.
Winnipeg's record against Edmonton this season dropped to 2-5-0.