In the sixth round, Kesler fired a shot to the stick-side of LaBarbera for a goal and Roberto Luongo made it hold up with a pad save on Boyd Gordon of Phoenix, giving the Northwest Division-leading Canucks a win to start a three-game homestand.
David Booth notched Vancouver's only regulation goal and Luongo finished with 23 saves.
Keith Yandle forced overtime with a goal from just inside the blue line with 2:06 remaining in regulation for Phoenix, which had its five-game winning streak snapped.
LaBarbera turned aside 21-of-22 Vancouver tries.
Notable deaths of 2014 [PHOTOS]