Backstrom improved to 6-1-0 in his last seven starts, blanking the Blues on 3-of-4 power play opportunities and stopping 12-of-13 shots during a busy second period for the Wild.
Cal Clutterbuck, Antti Miettinen and Kyle Brodziak all scored for Minnesota, which stayed in the midst of the Western Conference playoff hunt with its second win in two nights over struggling St. Louis.
The Blues got their only goal from Patrik Berglund en route to their fourth straight loss in St. Paul.
Jaroslav Halak started and was pulled in the second period after allowing three goals on 20 shots. Ty Conklin entered and went 11-of-11 the rest of the way.