I would look in these areas;
Suburbs: University City, Clayton, Richmond Heights, Brentwood, Maplewood
City neighborhoods; Central West End, Dogtown (Dogtown is the combination of three neighborhoods; Hi-Pointe, Clayton-Tamm, and Franz Park). I live in Franz Park and I am very happy with the area. As far as furnished, I'm not sure where to look.
You may want to check with this service, I believe it is free, and the money they make is actually from the landlords/property management companies;
CORT Relocation Services: Rental relocations and temporary assignments