Quick allowed six goals on 24 shots against Detroit earlier this week, so Bernier made the 27th appearance of his three-year career and he earned his third shutout.
Bernier's 25 saves allowed Los Angeles to jump into a tie with Chicago and Phoenix for fourth place in the tight Western Conference playoff race.
The game's lone goal was scored by Jarret Stoll with 7:43 to go in the third period while the Coyotes' Rostislav Klesla was in the penalty box for high sticking.
Phoenix lost its fifth in a row and suddenly is no longer a lock for the playoffs. Ilya Bryzgalov was the hard-luck loser, stopping 17-of-18 shots.