The Jets tied the game with a power-play goal midway through the middle frame, a tap-in by Vilardi after the original shot, which would have missed the net wide, instead hit Gudas' stick and laid flat in the crease for the former King.
Vilardi has collected six points in his last three games and sits third among team leaders in goals this season, his second as a Jet.
Winnipeg, the league's third-ranked offense, generated just eight shots on goal in the game's initial 40 minutes.
The Jets then claimed the lead early in the third though, as Scheifele buried a one-timer from Connor on a cross-seam pass through the Ducks defense.
The two-point night moved Connor into 11th in the NHL in scoring this season, passing Florida's Sam Reinhart and New Jersey's Jack Hughes.
Anaheim would respond with less than five minutes to play though, tying the game on a point shot by Vatrano that tipped off a Winnipeg defender and bounced on the ice before trickling through Comrie.