Shane Doan and Keith Yandle, on a power play in the final three minutes, also scored for Arizona (7-9-1). Goaltender Devan Dubnyk earned the Coyotes' first shutout of the season.
The Coyotes, who entered the game last in the Western Conference and 28th overall in the NHL, ended their losing streak at three games. Meanwhile, Dave Tippett recorded his 200th NHL career coaching win.
After entering the game tied for second overall, the Canucks (12-6-0) took only their third loss in the past 11 games.
Vancouver goaltender Ryan Miller allowed four goals on 13 shots in the first two periods as the Coyotes held leads of 1-0 and 4-0. Arizona scored the only goal of the opening session despite being outshot 12-3.
Dubnyk made 35 saves while the Coyotes managed only 23 shots. He earned the win in his first start since allowing five goals in a Nov. 2 victory over Washington. Dubnyk's last shutout came Dec. 27, 2013 while he was still with Edmonton, in a win over Calgary.