Paul Kariya, Steve Sullivan and Kimmo Timonen also scored for Nashville, which has won five straight overall and nine of 10. The Predators have scored at least three goals in eight straight games.
Pavol Demitra, Petteri Nummelin, Mark Parrish scored for Minnesota, which lost at home for the first time in 12 games dating to last season. It was also the Wild's first home loss to the Predators since Oct. 31, 2001 - a span of nine games.
With the game tied at 3, Weber took a pass from Sullivan and beat a screened Manny Fernandez from the slot 3:34 into the final period for his second goal.
Vokoun stopped 25 shots, improving to 7-0-1 in eight starts since allowing six goals to the Wild on Oct. 7. His biggest save was a left pad stop of Brian Rolston on a short-handed breakaway midway through the final period.
Nummelin gave Minnesota a 1-0 lead with a wrist shot from the top of the slot 2:58 into the first period. The goal was his first in the NHL since April 1, 2001, with Columbus. He played the last five years in the Swiss League.
Sullivan, who had three points in the teams' first meeting this season, tied it less than two minutes later when he beat Fernandez from low in the right circle.
Timonen scored on the power play at 16:57.
Minnesota tied it at two at 4:13 of the second when Demitra finished a 3-on-1 rush by beating Vokoun with a wrist shot high on the glove side.
Kariya and Parrish traded goals less than two minutes apart later in the period.
Notes: Nashville LW Martin Erat returned to the lineup after missing Wednesday's game with a bruised rib. He had one assist. ... Minnesota C Wes Walz returned after missing two games with a leg injury, but LW Derek Boogaard missed his second straight game with a lower body injury.