Get a bottle of contact lens saline solution. Set the bottle in a pan of warm, not hot, water to warm it up before use (the same way you would warm a baby bottle). Twice a day stream the solution into the cut to clean it out. Use a cottonball or clean cotton cloth to wipe away any residue. If your puppy has fur that is getting into the wound, trim it. You can use a little Neosporin on it or if you are worried about him rubbing his eye with his paw and licking it off, use tee tree oil applied with a cotton swab. Make sure you check him over to see if he got any puncture wounds as well. Those do need to be treated by a Veterinarian since they are deeper than a cut and will have trapped bacteria.
If your puppy's eye is still looking infected (oozing puss, bad odor, angry red around the wound) in a couple of days, you might want to seriously consider getting him into the vet for some antibiotics. Normally a cut on a dog is not a big deal but, what concerns me is that it is so close to his eye. The last thing you want is for him to have the infection spread to his eye
*I'm not a Veterinarian but, I play one on GFY. That and I used to be a Vet tech 