Many dogs, and cats for that matter, need constant cleaning around their eyes and/or mouth. Soft shop towels are fantastic to use to gently clean these areas. Just tear the amount required (no sense wasting a whole towel on a small animal), moisten with warm water, wipe.
Gentle for the pet and you can just throw it away. Makes sanitary sense to me!
Available at auto shops. Usually less expensive than the big stores that they are also available in.