Vladimir Sobotka and Alexander Steen also scored for the Blues, who improved to 14-5-1. Jake Allen made 29 saves.
Ryan Strome scored for the Oilers, who have lost three of their last four. Cam Talbot turned away 35 shots.
The Blues caught a break to start the scoring with 4:46 left in the first period. Talbot flubbed a clearing attempt, and it popped in front of the net, enabling Sobotka to get a short-handed goal.
Edmonton got the goal back on the same power play 29 seconds later when Strome's wrist shot beat Allen up high.
St. Louis regained the lead on the power play seven minutes into the second period when Steen whacked in a rebound during a scramble in the crease.
The Blues took a 3-1 lead six minutes into the third when Schenn elected to shoot on a 2-on-1 break, scoring his sixth goal of the season.
Schenn got his second goal of the game a couple of minutes later when he was left alone at the side of the net and had an open net to shoot at before Talbot could get across.
