Crouse tied it 1-1 on a power play at 10:20, chipping in a pass by Jason Zucker at the back post.
Matias Maccelli put Arizona ahead 2-1 on a power play at 15:24, converting on his second attempt after his initial shot was blocked by Lars Eller.
Carter tied it 2-2 with a backhand on a short-handed breakaway at 16:54, then put the Penguins up 3-2 when he deflected a point shot by Kris Letang on a power play at 4:23 of the third period.
“He’s just a really good pro,” Pittsburgh coach Mike Sullivan said. “I think he recognizes the circumstance, and he wants to do anything he can to help the team win. So, he’s willing to embrace any role we ask him to play. I think that’s a great example for all of us that we can learn from.”
Each of the first five goals in the game came on special teams.
"It's tough, especially for guys that aren't on the power play or PK,” Coyotes forward Logan Cooley said. “You're kind of sitting there while your legs get cold. But again, it's part of the game. It's a crazy first period, but that's no excuse for us.”
Guentzel scored an empty-net goal with 1:23 remaining for the 4-2 final.
NOTES: Carter’s 67th multigoal game of his NHL career tied Dallas Stars forward Joe Pavelski for fifth-most among active players. Carter also became the third active player with 20 career short-handed goals, joining Boston Bruins forward Brad Marchand (33) and Avalanche forward Andrew Cogliano (22). ... Pittsburgh defenseman Pierre-Olivier Joseph played 13:16 in his return from missing 16 games with a lower-body injury. … Cooley, a Pittsburgh native, took the opening face-off against Crosby in his first game in his hometown since being selected No. 3 by the Coyotes in the 2022 NHL Draft. At 5 years old in 2008-09, Cooley was an inaugural member of the “Little Penguins Learn to Play Hockey” program, an initiative spearheaded by Crosby to provide children with free equipment and on-ice instruction.