Brent Burns, Joe Pavelski and Timo Meier scored for the Sharks. Martin Jones allowed five goals on 19 shots and was replaced by Aaron Dell after the second period.
"We didn't play too well," Sharks forward Logan Couture said. "Disappointing. I think over 82 games in the season we had a pretty solid year to make the playoffs."
The Wild scored three unanswered goals in the second period to build a 5-2 lead. Granlund beat Jones with a wrist shot on a rush at 11:19 to make it 3-2. Zucker scored with a backhand at 11:47 on an odd-man rush to make it 4-2, and Zucker scored again at 15:25 on a 2-on-1 rush, beating Jones with a wrist shot for a 5-2 advantage.
Meier scored with 1:10 to play in the third period to cut the lead to 5-3, but Staal scored into an empty net with 26 seconds remaining for a 6-3 advantage.
"We were prepared," Dubnyk said. "The last three games, every game the other team had something to play for, so we knew it wasn't going to be a breeze through the last week of the season, and these are extremely good teams.
"Bruce [Boudreau] talked to us about it. It's true. It's not a switch, you can't turn it off and on, so we've been playing really good hockey and we wanted it to continue. We didn't want to have three games off and then try to get it back for playoffs."
Brodin scored on a slap shot from the left point at 10:53 of the first period to give Minnesota a 1-0 lead. Cullen made it 2-0 when he put a wrist shot from the low slot past Jones at 14:52.
Burns scored from the right point at 15:28 to make it 2-1, and Pavelski beat Dubnyk with a wrist shot from the slot at 16:32 for a 2-2 tie.
"Would have liked to play a little bit better at the start," Pavelski said. "But, we answered. We tie it up, we're playing, and then we let it slip."
