You don't have to be terrified about getting your chinchilla wet. Just don't soak him to the point where it messes with his body temperature which is the biggest hazard with getting chins wet. But many chins have survived the dreaded accidental dunk in the toilet. You can use water to bathe a chin if something really bad happens if you are careful to regulate his body temperature at all times until completely dried to prevent shock. You can even dunk half of your chinchilla in room-temperature water as a fast way to treat heat stress in an emergency.
Of course this is only a small greasy spot so you don't need to go that far. Try to only get the spot that needs cleaned damp. Chinchilla fur dries off just like any other kind of fur.
Using a washcloth should get most of the neosporin out. Since it's greasy you could try rubbing some dust into it first to make it easier to wipe off. It's better if you chin doesn't lick it off.