Suter tied it 3-3 at 19:24 with Demko pulled for the extra attacker on a goal-mouth scramble in the slot. Jonathan Lekkerimaki scored in the fourth round of the shootout to give Vancouver the win.
Demko, who missed 15 games, was activated from injured reserve and played for the first time since Feb. 8, when he sustained a lower-body injury midway through the first period of a 2-1 win against the Toronto Maple Leafs.
Lekkerimaki also scored in regulation for the Canucks (33-26-12), who are 2-2-1 in their past five games.
Vancouver pulled within three points of the St. Louis Blues for the second wild card into the Stanley Cup Playoffs from the Western Conference.
Timo Meier scored twice, and Jacob Markstrom made 25 saves for the Devils (37-28-7), who have lost three in a row and are 1-3-1 in their past five games.
New Jersey is seven points ahead of the New York Islanders and New York Rangers for third place in the Metropolitan Division.
Suter gave the Canucks a 1-0 lead at 10:09 of the first period on a one-timer from the slot off a pass by defenseman Marcus Pettersson from along the left-wing boards.
Meier tied it 1-1 with a power-play goal at 10:24 of the second period, jamming in his own rebound at the right post after a backhand pass by Dawson Mercer from the slot
Meier’s second goal gave New Jersey a 2-1 lead at 11:02 of the third period on a snap shot at the left post after an attempt by Brenden Dillon from the right circle banked off the end boards to Meier.
Lekkerimaki scored 45 seconds later to tie it 2-2 on a quick shot from the slot at 11:47.
Erik Haula gave the Devils a 3-2 lead at 13:25 on a backhand in the slot to extend his goal streak to three games (three goals).