Vladimir Tarasenko and Magnus Paajarvi also scored and Jaroslav Halak made 23 saves for the Blues, who have won 13 of 15 games against Buffalo since 1999.
Tyler Ennis scored the Sabres' goal just 10 seconds into the game, as Buffalo lost at home after consecutive home wins last week.
After a dump into the Blues zone off the opening faceoff, Ville Leino stripped the puck from Jay Bouwmeester below the left faceoff circle and found Ennis, who beat Halak with a low shot to give the Sabres an early lead.
Tarasenko scored a power-play goal with 36 seconds to play in the first period on a pass from T.J. Oshie, who collected his own blocked shot.
Morrow's second of the game came when Cody McCormick lost the puck behind the Buffalo net. Chris Stewart picked up the puck and passed to Morrow, who had raced into the high slot for a one-timer.
Paajarvi added his first goal for the Blues when he beat Miller below his blocker with just over six minutes played in the third.