Schenn got his team-leading 14th goal two days after netting a hat trick against Montreal. Colton Parayko and Vladimir Tarasenko also scored.
Allen got his first shutout of the season and the 16th of his career. He was briefly backed up by local vending machine worker Tyler Stewart after regular backup Carter Hutton was injured at the morning skate.
Dallas has lost two in a row since a season-high five game winning streak. Kari Lehtonen stopped 24 shots.
The Blues scored twice on the power play after connecting on just 2 of 17 chances in their previous four games. Schenn, acquired in an offseason trade with Philadelphia, pounced on a rebound from close range midway through the first period for a 1-0 lead.
Parayko blasted a shot from the just inside the blue line that eluded Lehtonen in the second period.
Tarasenko scored into an empty net with 47.6 seconds left, ending a six-game scoreless streak.
