Fixed gear drive chain injury

servicing my bike last night and got my finger sucked into my drivechain between the rear cog and chain, its usually the one thing im really careful about. Its cracked the nail and stabbed through abit of flesh, im worried the oil will make it infected and i ndont really wanna go to hospital if i can avoid it, anyone else done this and did it heal its self ok?

how do people manage to do this? Seems to be a really common injury