You might want to do something like this:
Add the user that you are logging into FTP as to a group, let's say "administrators."
# usermod -g administrators (username)
You can then modify the permissions of the "/var/www" directory to be writable by owner and group, like such:
# chmod 775 /var/www
And then finally, make "/var/www" part of the administrators group:
# chgrp administrators /var/www
This way the files that you upload should be uploaded as part of the administrators group, and apache will be able to serve the files just fine.