You can clean the pieces that way, yes. The core is metal, though, and water can cause problems with that. If you HAVE to clean the core, just wipe with paper towels.
Wipe it with paper towels. If that doesn't do enough, a drop or two of water while wiping with paper towels, will work. Just be sure to leave the core by itself for several hours in a ventilated area to ensure it dries out completely.