One word: Polysporin. The thing works miracles :)
I did that once by accident shaving, I soaked it in warm water & then wrapped a cloth around it because it was bleeding grusomely. Keep the cloth wrapped around it & against it firmly, do not remove it unless you have to wash it with anti-bacterial soap. Boy do I sound like a doctor ^_^
Just put some neosporin on it and then put a band aid on it. The neosporin will also help with preventing infection.
Could you use an electric razor?
Those would both make your cut burn. Put a piece of tissue or TP on the cut till it scabs over.