The game was tied 2-2 when Barron scored at 13:23 and Scheifele followed up with his team-leading 13th goal of the season at 15:12.
Saku Maenalanen and Nate Schmidt each had a goal and assist for the Jets (15-7-1). Dylan Samberg recorded his first NHL goal and Kyle Connor added a pair of assists.
Connor Hellebuyck made 28 saves for Winnipeg, which is 2-1-0 in a four-game homestand that wraps up Tuesday against the Florida Panthers.
Jakob Silfverberg and Brett Leason scored for the Ducks (6-17-3), whose winless streak stretched to six games (0-4-2).
Anthony Stolarz stopped 35 shots for Anaheim, which was coming off a 5-4 shootout loss to Minnesota on Saturday.
The Ducks led 1-0 after the first period and it was tied 2-2 following the second.
The Jets struggled to put pucks on the net in the opening period.
Winnipeg had the game’s first power play at 6:20, when the shots on goal were 5-0 in Anaheim’s favour, and the man advantage ended at 6-1.
The Ducks had a pair of power plays and made the second one count.
Hellebuyck stopped a shot by Frank Vatrano but the rebound bounced out to Silfverberg, who slapped the puck into the net at 15:16.
Winnipeg failed to capitalize on a second power play and the opening period ended with Anaheim outshooting the Jets 18-7.
The Ducks went up 2-0 after a Samberg giveaway near centre ice turned into a breakaway for Leason who converted the chance at 5:09 of the second.
The Jets responded with some pressure and were rewarded.
After Scheifele rang a shot off the post, Schmidt sent a puck from behind the net out front to Maenalanen and he put it behind Stolarz at 10:31.
Samberg redeemed himself for his earlier miscue when he tied the score 2-2 with a point shot through traffic at 13:28. Anaheim put in an unsuccessful coach’s challenge for a high stick.
The total shots on goal after the middle frame was 23-22 for Anaheim.
Schmidt finished the scoring with a power-play goal at 19:15.
Jets winger Blake Wheeler has been flying in his past five games, collecting 10 points (three goals and seven assists) … Rookie winger Michael Eyssimont was moved to Winnipeg’s top line with Dubois and Connor midway through the second period.
Jets: Host the Florida Panthers on Tuesday.
Ducks: Host the Carolina Hurricanes on Tuesday.