Kapanen scored his first two goals 2:31 apart to give the Penguins a 2-1 lead in the first period. He tied it 1-1 at 16:33 when Zucker passed out of a crowd along the far wall to the slot, where Kapanen tucked a shot around Talbot's left pad.
The second goal from Kapanen came on a one-timer, putting the Penguins ahead 2-1 at 19:04.
Kaprizov gave the Wild a 1-0 lead at 9:44. Alex Goligoski's shot went off the right post. The puck bounced out to Kaprizov, who backhanded it past a sprawling Jarry.
Guentzel scored in a second straight game by skating to the slot for a snap shot, making it 3-2 at 13:34 of the second period.
Spurgeon deflected a shot from Jonas Brodin, cutting it to 3-2 at 16:57 of the second.
Kapanen extended it 4-2 at 8:45 of the third period. After he shot off the left post, it hit Goligoski before crossing the goal line. The hat trick was Kapanen's first in 252 NHL games.
Spurgeon scored from the high slot to cut it to 4-3 with 3:19 remaining and Talbot pulled for the extra attacker.
Minnesota has trailed in six of its seven wins.
"It says we can't hold a lead, that's what it says," Wild coach Dean Evason said. "We have to understand that teams don't quit. This is the National Hockey League. ... But we were very resilient. The group just stayed the course."
NOTES: Hartman scored the latest game-tying goal in the NHL this season and the second in the final 10 seconds of a game (Torey Krug of the St. Louis Blues at 19:51 against the Los Angeles Kings on Wednesday). … Goligoski had an assist and three blocks in 20:15 of ice time in his return from missing three games with an upper-body injury. … Rust had two assists after being sidelined the previous seven games because of a lower-body injury. … Zucker had two assists in his first game against the Wild since being acquired in a trade from Minnesota on Feb. 10, 2020. … Kaprizov scored a goal to reach 60 points (29 goals, 31 assists) in 65 NHL games. He required the fewest games to get to 60 points among players to debut with the Wild, surpassing Marian Gaborik (93 games).