Paint or rub some Brush-Be-Gone on the leaves.

If you can get at the base of the vine, cut it back to there and put some brush killer over the surface of the cut, making sure that it gets all over the top (carefully, of course, as you don't want brush killer all over your hands or the rest of your garden). If it is one of the asiatic wisterias, with which I am fighting, it is an ongoing battle, and you've got to hit them as soon as you see them.

Here is a link that might be useful: Wisteria leaves

I recently attacked my Virginia Creeper with round-up. I took the tree branch nippers dipped them in a strong mix of round up and clipped down to the base as far as I could. Then took the round up mix and poured a small amount directly onto the creeper root stump. Once I know for certain it is dead I am going to have husband yank it out with a chain hooked to the truck. :)

Good Luck with your vine.
Tina Marie