Zdeno Chara also scored for Boston, which overcame a 2-0 deficit to earn a point in its 11th consecutive game, its longest stretch since 12 in row (8-0-2-2) Jan. 27-Feb. 21, 2004.
But it was Kelly's two goals that turned what was a poor start into another victory.
"Everyone wants to get more points," he said. "No one wants to be a defensive forward in the NHL."
Kelly is on a run of late, starting to mesh with linemate Rich Peverley, scoring six goals and collecting three assists in his past eight games. Kelly has nine goals on the season.
"He's doing the right things," said Peverley, who set up the first goal with a short-handed rush from his own end. "He's getting in the right position to score goals."
Boston coach Claude Julien also is impressed by Kelly's start.
"We knew he could provide us some," he said. "To this point he's probably exceeded what we expected from him."
Thomas won his eighth straight start as the Bruins rebounded from a shootout loss to Detroit on Friday.
Winnipeg's Ondrej Pavelec made 32 saves.
Trailing 2-1 early in the second, the Bruins had a 3-on-1 break, but Nathan Horton mishandled a pass on the wing and ended up drawing a tripping penalty a few seconds later. But Boston would tie it when Kelly scored a short-handed goal at 7:28.
Kelly scored after the rush by Peverley, who stole the puck from Zach Bogosian at the blue line, cut down the left wing and around Byfuglien before getting a shot on Pavelec. Kelly raced in and banged in the rebound.
The Bruins grabbed the lead 3-2 when Benoit Pouliot set up Kelly in the slot and he one-timed a shot past Pavelec.
"What hurt us was the rhythm of the game," Winnipeg coach Claude Noel said. "When you look at the second period, the rhythm of the game got changed."
Thomas made a pair of nice pad saves with about 3 minutes left, the first coming on Bryan Little's backhand bid from in close and Kane's rebound of the shot.
"It was just reaction," said Thomas, who had a season high saves. "The backhand pulled me wide and half of the net was still open. I've learned from experience that I leave the five-hole open when I try to cover the other half."
The Jets pulled Pavelec with 1:17 to play and Brad Marchand wrapped it up with an empty-net goal 8 seconds later.
The Jets grabbed a 2-0 lead on the goals by Kane and Byfuglien midway into the opening period. Byfuglien set up Kane on a 2-on-1 break, sending a pass to the slot where Kane shifted around Thomas and scored on a backhand.
Pavelec wasn't upset with how his teammates played -- just the results.
"They know how to play those close games. It's a very tough team to play against," Pavelec said. "We had a chance to win. We scored the first two goals.
"In the second we stopped playing at all and gave them the chance to come back and tie the game and overall it wasn't that bad a game -- on the road, against Boston, we did a pretty good job. It's all the little details and we got to be better."
Byfuglien scored on a slap shot from the top of the right circle, beating Thomas inside the far post.
Boston, which lost 3-2 to the Red Wings on Friday in a game that was filled with energy and was billed as a matchup of two of the league's best teams, looked a bit sluggish at the start.
After the two quick goals by the Jets, Julien called timeout and his team responded by opening things up a bit for the remainder of the period and ended up outshooting Winnipeg 15-12.
The Bruins cut it to 2-1 on Chara's power-play goal when the shot deflected past Pavelec off the stick of former Boston defenseman Mark Stuart 16:38 into the first.
It was Winnipeg center Nik Antropov's 700th career game. ... Boston LW Milan Lucic played his 300th. ... The Bruins have two short-handed goals this season, both by Kelly. ... Peverley collected his 100th career assist on Kelly's goal. ... The Jets closed a three-game road trip and open a five-game homestand Tuesday against Ottawa. ... The Bruins were 1-4 when they trailed by two or more goals, but they've won their past two, also a 4-3 shootout victory at Buffalo on Wednesday. ... Chara has 16 points (four goals, 12 assists) in his past 16 games. ... It was the 99th consecutive sellout at TD Garden.