The game appeared to be headed for overtime after Phoenix erased a 2-0 deficit in the third period on goals by newcomer Rostislav Klesla and Ray Whitney, the latter coming with 37.7 seconds remaining and the goalie pulled.
But Phoenix forward Radim Vrbata went off for hooking with 13 seconds to play. During the ensuing power play, Mike Ribeiro dug the puck out of the left corner to Loui Eriksson and he dished to Benn, who beat Ilya Bryzgalov on the glove side.
Kari Lehtonen made 25 saves for Dallas, which has won three straight after losing nine of its previous 10 games.
Bryzgalov stopped 27 shots for the Coyotes, who have dropped four in a row following an eight-game win streak.