Nick Foligno, Boone Jenner and Jack Johnson scored in regulation for the Blue Jackets. Jenner's score came on a penalty shot with 7:55 remaining in the third period. David Savard added two assists for Columbus, which has gone beyond regulation in four of their wins during their streak.
This was the teams' first meeting since the Penguins beat the Blue Jackets in six games in the first round of the playoffs last season.
"That's still in your mind," Jenner said. "Doing those handshakes after Game 6. It's always in your mind. It doesn't go away. To play them here the first time this year, the crowd was electric."
Kris Letang had two goals -- including the tying score with 11 seconds remaining in regulation -- and Blake Comeau also scored for the Penguins, and Thomas Greiss finished with 24 saves. Letang also scored in the shootout.
Pittsburgh was without star Sidney Crosby for a second straight game because of swelling to the right side of his face that team officials said doesn't appear to be mumps as they await results from additional tests.
The Penguins came in 10-1-1 in their last 12 regular season games against the Blue Jackets, including a 5-0 mark last season.
Letang's short-handed goal gave the Penguins a 2-1 lead at 7:22 of the third as he scored on a 2-on-1 break with a high flick shot over Bobrovsky.
"As the game went along, physically we met the challenge and we started to take over the game," Penguins coach Mike Johnston said. "Puck possession-wise through the last half of the game it was certainly in our favor."
Columbus countered with two goals in 5:44 to take the lead. After being hauled down on a break by Simon Depres, Jenner beat Greiss with a shot under his blocker on a penalty shot for his sixth goal of the season.
Then with the teams skating 4-on-4, Johnson followed with a shot from the left circle into the far corner of the net to make it 3-2 with 3:11 remaining.
Pittsburgh kept pushing down the stretch. With Greiss pulled for an extra attacker, Columbus twice was called for icing with under 25.5 seconds left bringing the face off back into the Blue Jackets' zone. Letang finally made took advantage with a one-timer from the left circle for his seventh goal.
"I think we played really well in the third period," Letang said. "We were after them and dictating the play. We deserved the point."
The Blue Jackets jumped ahead 1-0 on the power play at 3:02 of the first period partly because a series of odd bounces that put Pittsburgh flat-footed. Jenner got a step and fed a centering pass to Foligno, who stopped it with his skate before snapping home his team-best 14th into an open net.
Throughout the period there were hard checks all over the ice, culminating with two simultaneous fights with just under 5 minutes to go set off by Columbus' Brandon Dubinksy charging Steve Downie hard into the boards near the Pittsburgh bench. Veteran Jordan Leopold and Penguins call-up Bobby Farnham, playing his first NHL game, continued to punch as they fell and rolled along the ice. Downie and Dalton Prout scuffled near the glass.
"It was a hard fought win and it feels good," Jenner said.
Pittsburgh tied the score with 4:54 left in the period when Comeau ripped a high shot from the left circle past Bobrovsky for his 11th.
"We've got to be happy with the overall effort," Foligno said. "I thought everybody was outstanding. The fights, the emotion and the solid play again from Bob and then the timely goals. We're coming around and playing the right way."
NOTES: Both teams are dealing with a slew of injury issues. Columbus is the league leader in man-games lost and has six players out. Pittsburgh has seven players sidelined, headlined by Crosby. ... Todd Richards coached his 200th game for the Blue Jackets.