St. Louis has won eight of 11 overall. At 13-2-2, the Blues tied Boston, Pittsburgh and Minnesota for most home-ice wins.
Toronto lost its third in a row.
Brian Elliott, appearing in his third successive game, made 19 saves to improve to 6-1-1.
Steen scored his 22nd goal, second in the NHL to Washington's Alex Ovechkin, who has 26 goals.
Schwartz recorded the second three-point game of his career.
St. Louis scored three times in the first 16:10 to chase goalie James Reimer, who faced 15 shots. It was the seventh time this season the Blues have sent a starting goaltender to an early exit.
Backes got the first and last goals of the game. He beat Reimer with a 20-foot wrist shot at 4:48 of the first period. He scored into an empty net for his 15th goal and giving him 29 points, one more than his total for all of last season.
Schwartz and Roy scored 3:11 apart to push the lead to 3-0.
NOTES: St. Louis took 36 shots to Toronto's 22. The Maple Leafs have outshot their opponents only four times this season. ... The Blues are 18-1-2 when scoring first. ....Toronto D Paul Ranger played in his 300th NHL game. ... St. Louis has scored at least two goals in all 17 home games this season. ... Maple Leafs D Dion Phaneuf served the final game of a two-game suspension for a hit on Boston's Kevan Millar on Sunday. .... Toronto has not won in regulation in its last 12 games.