The Ducks (42-32-6) would have clinched their first Stanley Cup Playoff berth in eight seasons with a win of any kind. Anaheim, which has lost seven of eight (1-5-2), is third in the Pacific Division, three points ahead of the Los Angeles Kings.
Rossi scored with a one-timer from the right face-off circle with two seconds left on the power play after Chris Kreider was called for slashing.
Rossi also had an assist, Curtis Douglas scored his first NHL goal, and Jake DeBrusk and Brock Boeser also scored for the Canucks (24-48-8), who were coming off a 4-3 shootout win at the San Jose Sharks on Saturday. Nikita Tolopilo made 24 saves.
The Canucks have won two in a row for the second time this season and first since a four-game winning streak from Dec. 14-20.
Cutter Gauthier scored two goals after missing the previous five games because of an upper-body injury, Kreider had two assists for the Ducks. Lukas Dostal made 22 saves.
The Ducks still had the puck in the Vancouver zone after the Canucks killed the first power play of the game when Kreider tapped a rebound to Gauthier on the other side of the crease and he scored with a one-timer into the open side of the net for a 1-0 lead at 3:41 of the first period.
Douglas, a 6-foot-9 forward who was playing in his 41st NHL game after he was claimed off waivers from the Tampa Bay Lightning on March 6, pushed in a rebound that got behind Dostal in the crease to tie it 1-1 at 10:49.
Vancouver was on a power play when DeBrusk tipped in a point shot from Rossi for a 2-1 lead at 14:37 of the first period.
Boeser stole a pass back to defenseman John Carlson at the right point while the Ducks were on a power play and scored on a breakaway to make it 3-1 at 4:28 of the third period.
The power play continued and Gauthier scored his 40th goal of the season when he squeezed a one-timer through the pads of Tolopilo to cut it to 3-2 at 5:04.
Leo Carlsson tied it 3-3 at 6:56 when he flipped a backhand over Tolopilo from a sharp angle.
