Schaefer skated into the offensive zone on a 2-on-1 rush and put a wrist shot from the right face-off circle in off the left post.
Schaefer also had an assist, and Bo Horvat and Adam Pelech scored for the Islanders (7-4-2), who are 3-1-1 in their past five games. Ilya Sorokin made 22 saves.
Quinn Hughes had a goal and an assist, and Joel Eriksson Ek and Matt Boldy scored for the Wild (9-3-3), who had won three straight. Jesper Wallstedt made 22 saves.
Minnesota lost its first home game of the season (4-0-1).
Eriksson Ek made it 1-0 just 27 seconds into the first period with a wrist shot from the high slot.
Schaefer tied the game 1-1 at 16:36 with a wrist shot from the left circle on a rebound from Emil Heineman’s shot.
The Islanders took a 2-1 lead at 1:41 of the second period on Horvat’s backdoor tip-in from Simon Holmstrom on the power play.
Schaefer had the secondary assist to become the sixth teenage defenseman in NHL history with 15 career multipoint games. The others are Phil Housley (34), Ray Bourque (17), Larry Murphy (16), Rick Hampton (16) and Rasmus Dahlin (15).
Hughes tied it 2-2 after he cut through the right circle and beat Sorokin five-hole with a sharp-angle, backhand shot at 17:21.
Pelech put New York ahead 3-2 at 7:14 of the third period when his wrist shot from the point deflected off the left leg of Minnesota forward Michael McCarron and changed direction to get past Wallstedt.
Boldy evened the score 3-3 at 16:07 with a low snap shot between Sorokin's pads from the left circle.
Islanders defenseman Ryan Pulock left the game early in the first period with an upper-body injury after just two shifts.
