Quick turned aside all 30 shots he saw, including 14 in the second period.
Slava Voynov scored the game's only goal 5 minutes into the middle frame.
The LA goaltender came up with the standout performance as the Kings were facing a 2-0 deficit in their Western Conference quarterfinal series.
Los Angeles, which outhit St. Louis 53-39 in the physical contest, will attempt to even the series when it hosts Monday's Game 4.
Brian Elliott turned aside 20 shots for the Blues.