Winnipeg's backup goalie got the start against the team that drafted him and only faced four shots in the first period, but later made some key saves on the way to the Jets' 2-1 victory.
Mathieu Perreault scored the winning goal with 23 seconds left in the second period, a tipped shot that was originally credited to Dustin Byfuglien.
"I get energy from watching the guys work so hard in the other zone. They really gave it to them," Hutchinson said of Winnipeg's 16-4 shot advantage in the opening period.
"It's easy to stay in the game when you're just enjoying your team battle in the offensive zone."
Hutchinson, who was drafted by Boston in 2008 (77th overall), made 30 saves in Winnipeg's second-straight win (17-10-6).
Evander Kane had Winnipeg's other goal.
Tuukka Rask made 25 saves for the Bruins (16-14-3) with Torey Krug scoring Boston's lone goal.
Defenceman Jay Harrison, acquired by the Jets in a trade with Carolina on Thursday, introduced himself to fans with an assist in his first shift.
Harrison made a pass from the point over to defenceman Grant Clitsome, whose straight-on blast hit Rask. Kane then zoomed in and flipped the loose puck into the net at 3:12 for his sixth goal of the season.
Rask said he threw his blocker out on Kane's shot, but didn't get enough.
"Right now, I can't afford to make those mistakes the way we've been scoring goals, but it didn't decide a game or anything," he said.
Boston tied the game 1-1 at 4:49 of the second period after Chris Kelly cut across the front of the net and sent a backhand shot at Hutchinson. Krug got the rebound and scored his sixth of the season.
Byfuglien's wrist shot from the point went through traffic, including four players just outside the crease, and was deflected by Perreault for the 2-1 lead at 19:37.
"I saw (Byfuglien) had the puck. Then they started cheering and I turned back and I didn't even see the puck in the net, so no chance there," Rask said.
The goal extended Perreault's point streak to six games with two goals and four assists, the team's longest this season.
Mark Scheifele also assisted on the goal, giving him seven points in his last four games, including three goals.
Boston then outshot the Jets 11-7 in the second period.
The Bruins head coach Claude Julien said his players didn't have a good start, but that wasn't the reason for the loss.
"The biggest thing for me is that we passed up on so many shot opportunities and looked for the cute plays," he said. "That's not our team."
Having the Jets score so late in the second didn't help either.
"We don't play through players and they keep the puck in the middle of the ice. They put it back in and then we don't box out and it's in our net," Julien said.
"Those kind of mistakes. We're making costly mistakes."
The teams each took a penalty within a minute of each other midway through the third period, but neither could find the back of the net during their advantage. Hutchinson made two good saves when Boston had the extra man.
The Bruins pulled Rask with just under two minutes left, but Hutchinson withstood the attack.
Harrison played 20:18 with his new team, even though he didn't arrive in Winnipeg in time for the morning skate.
"A lot of times, the more times you go over the boards, the easier it gets because you get into a nice routine and rhythm and there's less thinking and more playing," said Harrison, who has played more than 300 NHL games.
The Jets end a four-game homestand Sunday against Philadelphia. Boston returns home to host Buffalo on Sunday.