A bad line change by Winnipeg proved costly as Hadyn Fluery's attempted pass to Vladislav Namestnikov, who was changing, was picked off along the boards by Shea Theodore, who moved in on a 2-on-1 before dishing to Barbashev who fired the disc past a shocked Hellebuyck.
Keegan Kolesar and Victor Olofsson also scored for the Golden Knights, who kicked off a three-game road trip with a win.
Shea Theodore had a pair of assists.
Adin Hill turned aside 18 shots for Vegas, who moved to 19-7-3.
Nikita Chibrikov also scored for the Jets.
Connor Hellebuyck made 33 saves for Winnipeg, who dropped to 21-9-1.
Chibrikov gave the Jets the lead 9:27 into the first when he drove the net and shot a rebound off a Vladislav Namestnikov shot into an almost wide open net.
The goal was his 2nd of the season and third in his three NHL appearances.
There was no scoring in the second period.
Winnipeg didn't register their first shot on goal of the second period until the 18 minute mark.
Keegan Kolesar scored 4:34 into the third when he parlayed a solid shift of forechecking into the tying goal.
"It's fun, it's very fun. Got a lot of special people in the building," said Kolesar, who grew up in Winnipeg. "It's exciting." Logan Stanley attempted to get a pass to Kyle Connor, but the disc ended up hopping over his stick and Kolesar took possesstion before firing home a wrister past Hellebuyck from inside the left circle to make it 1-1.
The goal was his 7th of the season.
Josh Morrissey scored at 10:55 and gave the Jets a one goal lead.
Morrissey took a partner pass along the Vegas blue line before moving in to the top of the face off circle and blew a slapshot past Adin Hill, beating him over his right shoulder.
The goal was his 3rd of the season.
Victor Olofsson tied it up with a power play marker at 18:11 when Mark Stone hit him with a cross-crease pass which the Vegas winger one timed past Hellebuyck.
The goal was his 4th of the season.
"I mean I think we played a solid game, I think we were the better team and we just kind of kept playing the same way the entire game and we finally got paid for it at the end," said Olofsson.
The game went into overtime.
"We were down two to one in the third and I thought we were playing really well to be honest with you especially defensively. They (Jets) are at the end of I think a four in six, they've had a tough stretch, we've been off for a while so I thought we might be rusty early," said Vegas Head Coach Bruce Cassidy.
"I thought after the first period that we executed better and had our legs and hands more than I would have suspected. Didn't give up much for two periods but hung in there and got some calls late we needed them obviously and our power play was able to connect there.
Both teams are active Saturday as Vegas visits Edmonton and Winnipeg hosts Montreal.
SHOTS-VEGAS 36 WINNIPEG 20
ATTENDANCE-13,320
