Evgeni Malkin, Phil Kessel and Justin Schultz each had a goal and an assist, and Matt Murray made 29 saves for the Penguins, who fell behind 1:40 into the first period.
"The building was loud," Pittsburgh forward Jake Guentzel said. "It's their home ice and we knew they were going to come out strong. Not the way we wanted to start for sure. We have to start better next game and go from there."
The winning goal was reviewed for goaltender interference, but the call on the ice stood after officials determined that Penguins defenseman Kris Letang caused Barzal to make contact with Murray.
Bailey had a chance to score the game-winner late in the third period but hit the post with seven seconds remaining.
"That was tough to shake," he said. "I was worried I was going to have to see that one in my nightmares for a while. Little bit of redemption. I'm just happy to get the win. It's a crucial game, but there's still a long way to go."
Schultz tied the game 3-3 with 1:29 left in the third period with Murray pulled for an extra attacker.
"It's a tough building to play in," Schultz said. "The fans are on top of you. I liked how we battled back, but still lots to learn. … The guys have been here before, and we know it is going to be a long series."