In the past when I have had to do this, I ended up manually modifying bigip.conf and moving the specific configuration into the new partition. I'm not sure there is a way to handle this via the UI. I wish there was, though.
In the GUI, you'd select the partition from the top left and then create the virtual server. You should be able to edit the bigip.conf to move existing virtuals to a new partition. But reloading the config could be disruptive to existing connections.