Evander Kane had Winnipeg's other goal while Michael Hutchinson, who was a Boston draft pick in 2008 (77th overall), stopped 30 of the 31 shots he faced for the win.
Tuukka Rask made 25 saves for the Bruins (16-14-3). Torey Krug scored Boston's lone goal.
Defenceman Jay Harrison, acquired by the Jets in a trade with Carolina on Thursday, introduced himself to fans at MTS Centre with an assist in his first shift.
The Jets (17-10-6) led 1-0 after the first period and took a 2-1 advantage into the third period after Perrault's goal at 19:37 of the second.
Winnipeg had a pair of power plays within the game's first four minutes, but both their goals came at even strength.
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.
After Boston's penalties expired, Winnipeg was outshooting the Bruins 9-3.
The Bruins had a power play in the opening period, but didn't register a shot on net. They would only get one more as Winnipeg had the 16-4 shot advantage after the period.
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. Scheifele also assisted on the goal, giving him seven points in his last four games, including a trio of goals.
Boston outshot the Jets 11-7 in the middle period.
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.
The Jets end a four-game homestand Sunday against Philadelphia. Boston returns home to host Buffalo on Sunday.