Nik Antropov scored for Winnipeg (2-5-1) in front of a soldout crowd of 15,004 at MTS Centre.
The Rangers (3-2-2) began the preseason in Sweden and stayed on the road. Their home opener is Thursday against Toronto in the renovated Madison Square Garden.
Biron was in goal instead of star Henrik Lundqvist as part of a planned rotation. Lundqvist left Saturday's 2-0 loss to Edmonton in the third period because of pain from new skates.
The Jets now embark on their own seven-game trip, beginning in Philadelphia on Thursday.
Winnipeg came out strong in the third, outshooting the Rangers 12-1 over the first eight minutes until Jets forward Chris Thorburn was sent to the box for tripping, leading to Callahan's winner.
"I don't think our players can sit there and hang their heads on the effort they gave," Jets coach Claude Noel said.
"I think we probably deserved a better fate. Didn't get it, so we move on to the next game. ... We did everything right but win the game."
Chris Mason got the start and made 15 saves for Winnipeg. On Saturday night, Mason replaced Ondrej Pavelec after the young netminder gave up two early goals against Carolina and the Jets went on to a 5-3 victory.
"I think we've had some other losses where we've probably been pretty disappointed in the way we played," Mason said.
"It just came down to they kept the puck out of the net in the third period and we had an unfortunate bounce."
Callahan scored after his shot went off the skate of Winnipeg defenseman Zach Bogosian at 8:41.
New York went 2 for 5 on the power play and the Jets were 0 for 1.
Antropov said the lengthy road trip may be good for the team.
"It's going to be a lot of team bonding and stuff like that," Antropov said.
"It's a good thing to get together as a group and spend time with just the guys."
Alex Burmistrov assisted on Antropov's goal and is now on a five-game point streak. ... Jets center Jim Slater sat out after taking a knee to the head in Saturday's game and is listed day to day.
Copyright by STATS LLC and The Associated Press
Arena: MTS Centre Location: Winnipeg, Manitoba
Referees: Mike Leggo, Mike Leggo Linesmen: Steve Barton, Mark Shewchyk
Attendance: 15,004 (100.0% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
No scoring this period
Ruslan Fedotenko (1)
(Power Play) Assists: Dan Girardi, Michael Del Zotto
Nik Antropov (2)
Assists: Alex Burmistrov, Evander Kane
Ryan Callahan (2)
(Power Play) Assists: Ruslan Fedotenko, Dan Girardi