Backes' first tally gave the Blues a 3-1 lead midway through the final frame as he tapped in a T.J. Oshie shot that slowly trickled behind Sabres goalie Ryan Miller.
The Blues captain added an empty-netter while B.J. Crombeen and David Perron also scored for St. Louis.
Jaroslav Halak earned the victory with 19 stops.
Buffalo continued its futility away from home, adding to a club-record 12-game road losing skid.
Tyler Myers and Mike Weber scored while Miller yielded three goals on 26 shots in the Sabres' loss.