Evgenii Dadonov had a goal and an assist, and Robin Lehner made 32 saves for the Golden Knights (23-15-2), who are 1-3-2 in their past six games, all of which have come at home.
"You don't want to lose at home," Vegas captain Mark Stone said. "We pride ourselves on being a good home team. It's been a tough stretch for us against good teams, but you've got to find a way to get points."
Trailing 3-0 after the first period, Zucker cut it to 3-1 at 6:56 of the second with a power-play goal, jamming in a rebound near the right post.
Teddy Blueger then scored 49 seconds later off a rebound to cut it to 3-2.
Pittsburgh had the first 13 shots of the second period.
"Down 3-0 at the end of a long road trip, it would have been easy to start feeling sorry for ourselves and pack it in," Penguins defenseman Mike Matheson said. "We did the complete opposite."
Zucker tied it 3-3 when Kapanen's shot off a face-off win by Evgeni Malkin deflected in off his stick 28 seconds into the third period.
"I don't think our starts these past three games or so have been our best," Kapanen. "It's something we need to address, but just kind of trying to get the things moving again and just believing in our game."