Elliott racked up his third straight win and his first shutout of the year. He was busiest in the second period, when he stopped 13 Detroit tries.
Chris Porter netted the only goal of the contest late in the second period, sending the surging Blues to their fourth straight victory.
St. Louis came into the contest with a two-point lead in the race for the eighth and final Western Conference playoff spot.
Jimmy Howard made 24 saves for the Red Wings, who hold the West's seventh seed.