On the winning play, Brad Richardson got control of the puck on the left side and sent a pass into the slot, which found Bieksa streaking toward the net.
The Canucks defenseman shoveled a shot at Mike Smith, who blocked the first attempt, but Bieksa poked in the rebound while falling to the ice, sending Vancouver to its third victory in four games.
Roberto Luongo earned the victory with 29 saves.
Antoine Vermette recorded his third career hat trick and second in 36 days for the Coyotes, who came into the game trailing the Canucks by four points in the Western Conference wild card race.
Shane Doan notched a goal and an assist and Smith finished with 20 stops.