Barron scored at 13:23 for a 3-2 lead. Scheifele followed up with his team-leading 13th goal of the season at 15:12.
The Jets were outshot 18-7 in the first period and trailed 2-0 early in the second.
''It's just not right to start a game like that,'' Winnipeg's Rick Bowness said. ''Listen, I'm the head coach so I'm responsible for the way we start and getting our team prepared. So that starts with me. The second thing is, I'm not a babysitter. These guys are men. They're professionals and they're paid to show up here and go to work. My job here is to make that happen.''
''The third thing on that would be: You cannot play this game without passion. Without emotion. You cannot play this game on your heels. And I hate when we're on our heels. It's not right.''
Saku Maenalanen and Nate Schmidt each had a goal and assist for the Jets. 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, 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.
Barron said the players absorbed Bowness' words in the dressing room after the first period.
''There wasn't enough intention. It wasn't intense enough from us,'' Barron said. ''We weren't playing with enough passion in that first period. We all knew it on the bench, on the ice, and I'm sure you guys could tell from the press box that was the case. We need to have better starts.''
Silfverberg scored on a rebound on a power play late in the first period. The Ducks went up 2-0 after a giveaway by Samberg near center ice turned into a breakaway for Leason at 5:09 of the second.
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 of the middle period.
Samberg redeemed himself for his earlier miscue when he tied it at 2 with a point shot through traffic at 13:28. Anaheim put in an unsuccessful coach's challenge for a high stick.
''We had a really good first period and we let them get back in the second, and the third period was anyone's game and they jumped on the chance,'' Leason said. ''We just got to find a way to get good starts and finish games.''
Jets winger Blake Wheeler has three goals and seven assists in his last five games. . Rookie 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.
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