Cats keep their fur clean but the saliva is what causes allergic reactions in humans. The dried saliva falls off with the cat's dander, tiny dry flakes of skin that shed naturally just like they do in humans. In order to keep the dander down, bathing a cat is one option.
First you have to fill a sink or other container with warm (not hot) water. Don't put the cat into a sink with running water as the sound of the faucet may frighten them. A friend recommended putting a screen inside the container, leaned against the side so the cat will have something besides your body to curl their claws around.
Hold the cat gently and pour cups of the water over him. You can use a mild shampoo or special pet shampoo. When you are finished, wrap your cat (including their claws) in a dry towel and rub them to take most of the water from their fur. Most cats won't tolerate a blow dryer, but if yours does, put it on cool or warm (be careful not to get it too hot) and blow the fur dry. Otherwise, just rub out as much moisture as you can and let your cat go.
He may be mad and not speak to you for a few days, but he'll get over it. And with any luck, so will you.