Brock Nelson and Anders Lee also scored during a three-goal second period for New York, which won for the first time this season. Thomas Greiss made 35 saves, including 15 in the first period.
"That's the way we need to play; that's kind of our identity right there," Bailey said. "We were playing hard all over the ice, tight defensively, but we were still creating a lot and found a way to hit the back of the net a couple of times too."
WATCH: [All Jets vs. Islanders highlights]
Patrik Laine scored, and Laurent Brossoit made 29 saves for Winnipeg, which has lost two of three on a season-opening four-game road trip.
Brossoit made a glove save on Leo Komarov's shorthanded breakaway at 13:10 of the first period and then back-to-back pad saves against Nelson on a breakaway during a Jets power play at 11:15 of the second.
"Sadly, he was fantastic," Winnipeg coach Paul Maurice said Brossoit, who made his second straight start. "I mean, the number, when you play as well as he played, you deserve a better number and he should be sitting the locker room having a good feeling right now. But boy, he's been like that right from the start of [training] camp. He's just been right-on for us.
"We got in trouble there in the second period, obviously. He was composed, made some great saves and gave us a chance to hang around a little bit."
Bailey gave the Islanders a 1-0 lead at 1:51 of the second period when he redirected Beauvillier's pass from the right circle for New York's first power-play goal of the season.
Nelson made it 2-0 when he took a pass from Bailey and put a wrist shot past Brossoit at 4:38 of the second.