Sidney Crosby scored the winner after Evgeni Malkin scored on a penalty shot of his own, and Marc-Andre Fleury made saves off Andrew Ladd and Evander Kane to secure a sixth consecutive win for the Penguins.
Blake Wheeler scored for the Jets to open the shootout.
Crosby, who along with Malkin was held off the score sheet through 65 minutes, called the game "out of the ordinary."
"It was two teams that don't see each other that much," Crosby said. "I think they wanted to come out hard and play well in front of their crowd, and I think we handled it well."
Brandon Sutter had a goal and two assists, while Steve Downie and Simon Despres added goals for Pittsburgh (9-2-1) in regulation time.
Ladd, Jacob Trouba and Evander Kane replied for the Jets (7-5-2), who now have points in seven straight games.
"I think we just capitalized on our chances," Sutter said. "I think overall we could still play a little bit better, but that was a fun game. That was two good teams going at it.
"You get into a shootout and we've got a couple guys that know a thing or two about scoring goals," he added.
The teams combined for 102 penalty minutes and three fights, but no power-play goals.
The Jets entered the game with the fourth-best penalty kill in the NHL, and managed to shut down the league's top power play. The Penguins, meanwhile, extended their perfect stretch on the penalty kill to 34 for 34.
Winnipeg had a chance to win the game on the power play after Despres was called for tripping Mark Scheifele with 1:06 left in overtime, but couldn't find the net. Each team was also given a power play in the final five minutes of regulation time.
Ondrej Pavelec made 32 saves for the Jets, while Fleury stopped 34 shots for the Penguins.
"It was a good battle," said Jets coach Paul Maurice. "It was fun to be a part of.
"We were ready to play from the start, played hard, made some mistakes, forced some mistakes… everyone in there is tired, and they should be."
Ladd, the Winnipeg captain, opened the scoring midway through the first period when he tipped a Bryan Little shot in the top of the crease past Fleury for his sixth goal of the season.
The goal ended a shutout streak of 164:22 for Fleury, who was playing his first game since signing a four-year contract extension worth $23 million on Wednesday.
The lead lasted 100 seconds, as Despres' point shot through heavy traffic made it 1-1.
Sutter gave Pittsburgh the lead at 6:50 of the second period when he broke through the Jets defence and beat Pavelec with a low shot for his fourth goal of the season.
Downie made it 3-1 late in the second when he just missed a breakaway pass after coming out of the penalty box, but took the ricochet off the end boards and made a deke to his backhand. Jets defenceman Dustin Byfuglien cross-checked Downie into the corner boards after the goal, but Downie's retaliation cost Pittsburgh a power play.
An undisciplined second period could have gone even worse for the Jets, but they managed to kill two power plays — including a 29-second two-man advantage. The Penguins' power play entered the game a league-best 41.3 per cent, but finished the night 0 for 6.
A blast to the top corner from Trouba late in the period made it 3-2 at the second intermission.
Kane tied the game on a breakaway at 3:55 of the third period after a lead pass from Wheeler. It was the first of the season, and 100th of his career, for the 23-year-old winger who missed eight games with injury.
"We wanted to come out and prove we could play with the best of the best," Kane said. "It was a very emotional game… old school hockey is the reason why you want to play the game."
The Jets begin a five-game road trip in Ottawa while the Penguins continue their five-game trip in Buffalo, both on Saturday.
Notes: Little played his 500th career game, while Penguins forward Craig Adams played his 300th consecutive game. … Wheeler, who entered the season with three fights in 454 career games, dropped his gloves for the third time this season in a first-period scrap with Rob Bortuzzo.