A vampire can outsmart a zombie easily. A vampire can fly. A zombie cannot. A vampire is ALOT faster than a zombie. (unless it's like a zombie type thing from 'I Am Legend' lol, very unlikely though). Here, let me do it like this:
Strength>Zombie Intelligence>Vampire Stamina>Vampire Wits>Vampire Better Characteristics (ability to fly, swim, fight, etc. Stuff like that)>Vampire
Vampire easily. If it doesn't have any rivers around it, it would know how to kill a zombie. It will turn into a bat, go to the nearest gun store and get either a shotgun or a springfield. Heck, if it wants to have some fun, AK 47.
I'm pointing in a direction. It would follow that direction. The vampire would go bat then and hide in a dark corner. Then strike when the zombie gets out and it sees a usable weapon.