Get some Shooters Choice solvent, a Dewey's coated rod, and a newspaper or something you don't mind getting all sorts of cruddy. Get a couple 22 caliber bore brushes, as this process will kinda destroy them.
Put a brush on the end of the Dewey's rod, wrap a bit of patch around the brush, then from the breach of the Savage, with the muzzle resting on the newspaper, run the brush all the way down to the muzzle and back up. Repeat that motion until no more cruddy solvent is forced out the bore. Change patches around the brush and using more solvent, scrub the bore again. Keep the muzzle down, as gravity helps keep the cruddy solvent from coming out of the chamber into the rest of your action.
Do this three to five times. What you are doing is actually scrubbing the inside of your bore with enough solvent to really give it a good cleaning. I've never had a bore not be totally clean by five patches using this method.
Switch to a piercing jag, then push clean patches through from breach to muzzle until they come out dry and clean. Run an oil patch down the bore and you are done.
Then start reloading, and switch to CFE 223 :-)