Although generally used with sprains or pulls, you can also use the RICE method here. This would refer to Rest, Ice, Compression, and Elevate. You would need to first take care of the actual cut. First apply pressure until the bleeding stops.
Next, clean the area thoroughly with soap and water. Add ointment and a snug fitting bandage to protect the cut. Rest your hand and do not use it as much as possible. This will also help to reduce the swelling. The next step would be to apply ice to the affected area and elevate it to eliminate any swelling.