St. Louis has won 10 straight games against Arizona, outscoring the Coyotes 42-13 in that time. The Blues have won 10 of their last 12 overall and trail Nashville by a point for third place in the Central Division.
Arizona has lost six of its last seven. The teams meet again Wednesday in Phoenix.
Alex Pietrangelo and Vladimir Tarasenko also scored for St. Louis, which is 16-1-1 against Arizona since 2011-12.
Allen improved to 7-0 lifetime against Arizona. He has won eight of his last 10 starts, allowing just 13 goals.
Arizona goalie Mike Smith stopped 24 shots.
