Erik Karlsson cut it to 6-4 with a slap shot at 3:01 before Couture tapped in a pass from Meier to make it 6-5 at 4:09.
"You never know what's going to happen," Karlsson said. "If you would've told us that it would be a one-goal game with two and a half, three minutes left in the third, I don't think anyone would believe you. But I think once that first period was over with, we just kept a short memory and we tried to do everything we could to establish our game moving forward."
Rodrigues completed his first NHL hat trick on a power play to make it 7-5 with 2:01 remaining before Rust scored his third of the game for his fifth NHL hat trick on an empty-net goal with 11 seconds left for the 8-5 final.
"We obviously got off to that hot start," Rodrigues said. "Everyone was kind of feeling good, then we got up 6-1. It's kind of a mental battle, trying to stay engaged. In your head, you know you have to. But sometimes, it's just natural to let your foot off the gas a little bit."
NOTES: Rodrigues and Rust are the second set of teammates in the past 10 years to have a hat trick in the same game (Ryan Nugent-Hopkins and Leon Draisaitl in the Edmonton Oilers' 8-4 win against the Los Angeles Kings on March 26, 2019). … The Penguins' six first-period goals are tied for the most in their history (Feb. 21, 1976, against the Chicago Blackhawks). … Guentzel became the second Penguins player in the past 10 years to have at least one point in 14 straight appearances (Pascal Dupuis, 17 games in 2011-12). Guentzel scored 19 points (12 goals, seven assists) on an NHL career-long 13-game point streak before being injured in a 6-1 win against the Seattle Kraken on Dec. 6. … Pittsburgh forwards Sidney Crosby and Danton Heinen, and defensemen Kris Letang and Marcus Pettersson each had two assists. … Sharks forward Rudolfs Balcers (lower body) will likely miss at least another week, Boughner said. He has been out since Dec. 3.