Jonathan Lekkerimaki scored his first NHL goal, and Kevin Lankinen made 27 saves for Canucks (8-4-3), who have lost two of their past three games after going 7-1-1 in their previous nine.
Dakota Joshua made his season debut after being diagnosed with testicular cancer over the summer. He was minus-1 in 12:11 of ice time.
Lekkerimaki, who was playing in his second NHL game, put the Canucks ahead 1-0 at 2:40 of the first period. He scored with a one-timer off the rush below the blocker of Varlamov from the top of the right face-off circle.
Jean-Gabriel Pageau tied it 1-1 with a power-play goal at 17:23. Lee collected a hard pass from Oliver Wahlstrom at the left post before sending a no-look backhand feed through the crease to Pageau for a tap-in.
Mayfield put New York ahead 2-1 just 14 seconds into the second period when his shot from the left point deflected in off the sticks of Canucks forwards Pius Suter and J.T. Miller.
Engvall extended the lead to 3-1 at 2:10, jamming in a rebound from the top of the crease after Lankinen made a good save off Simon Holmstrom from the slot.
Kyle Palmieri appeared to make it 4-1 at 8:36 of the third period following a turnover in front of the net, but the Canucks successfully challenged the play for offside.
However, Lee did make it 4-1 at 11:42. Horvat beat Quinn Hughes wide down the right wing and passed back out front from below the goal line to Lee, who was left alone for a quick shot from the low slot.
Noah Dobson scored his first goal of the season into an empty net at 17:54 to make it 5-1. Tyler Myers scored on a wrist shot from the right circle at 18:34 for the 5-2 final.