Kaprizov, who missed the past 28 games with a lower-body injury, ended it with a one-timer from the right circle off a cross-ice pass from Mats Zuccarello.
Joel Eriksson Ek had four goals in his first game back after missing 21 with a lower-body injury, and Matt Boldy had four assists for Minnesota (43-29-7), which holds the tiebreaker over the St. Louis Blues for the first wild card into the Stanley Cup Playoffs from the Western Conference.
Marc-Andre Fleury, who will retire at the end of the season, made 24 saves in what was likely the final regular-season home start of his NHL career.
Macklin Celebrini scored his first NHL hat trick and had two assists for San Jose (20-47-11), which lost a seventh consecutive game (0-5-2). Will Smith had a goal and three assists, and Alexandar Georgiev made 36 saves.
Tyler Toffoli gave the Sharks a 1-0 lead at 12:11 of the first period with a wrist shot in the slot that deflected past Fleury off Marco Rossi’s stick.
Marcus Johansson tied it 1-1 at 14:16, curling to the top of the left circle to take a drop pass from Frederick Gaudreau and scoring with a wrist shot from the high slot.
Brock Faber put Minnesota in front 2-1 at 18:48 with a snap shot from the high slot off a drop pass from Boldy.
The teams then combined for three goals in 1:06, and six total goals in the second period.
Celebrini tied it 2-2 at 7:29, losing Faber in front to score from the top of the crease, but Eriksson Ek put Minnesota back on top 3-2 at 7:47 when he scored on his third effort during a goalmouth scramble.
Carl Grundstrom tied it 3-3 at 8:35, jamming in a loose puck in the crease after Fleury knocked down Cameron Lund’s shot.
Celebrini put the Sharks ahead 4-3 at 12:28 with a one-timer in front.
Kaprizov responded to tie it 4-4 at 17:02 with a power-play goal, taking a cross-ice pass from Zuccarello and scoring with a one-timer from a tight angle at the bottom of the right circle.
Eriksson Ek made it 5-4 on the power play at 19:48, burying a loose puck in the crease.
Eriksson Ek completed his hattrick to make it 6-4 at 1:02 of third period, again on the power play, when he found a rebound at the top of the crease and chipped it past Georgiev. He added his fourth goal to extend the lead to 7-4 at 2:06 with a one-timer in the slot.
Nikolai Kovalenko brought it to 7-5 at 4:44 after Celebrini found him cutting to the slot.
Celebrini cut it to 7-6 at 10:00 with a backdoor tap in off a wraparound pass from Smith, who then beat Fleury with a wrist shot from the top of the right circle to tie it 7-7 at 19:08 with Georgiev on the bench for an extra attacker.
