Dog accidentally gouged my thumb; it HURTS, is bleeding for 4 mins?

Yes it was a true accident; I had a treat between my thumb and finger, letting her lick and gnaw on it to keep her attention, and she got my thumb between her bicuspids. So I have a kind of puncture wound, no bigger than a pencil eraser, on the side end of my thumb. It hurts terribly and is issuing a steady blood flow (would this be venous?). I applied pressure for a minute, it's still bleeding. I wrapped a band-aid on it after rinsing it tightly, I can see a stain forming under it. Do I pressure to an artery next? If so, where? I'm elevating it above my heart. Could it be my nail hurting, since it looks cracked? What do I do with a broken nail?