Without pictures it would be hard to identify any vine, but I would recommend you purchase a vine killer from your local nursery or garden dept--one that has Triclopyr is particularly effective for me (like Ortho Brush-B-Gone).

Here is a link that might be useful: Ortho Brush B Gone

grnthumber,
So, how would you use this product without killing the bushes too?

Very carefully! I have weed trees (from our neighbors' locusts and silver maples) growing next to most of my roses and shrubbery--yes, it is very frustrating!--I just carefully spray the leaves of the trees I'm wanting to kill while avoiding as much as possible the leaves of the desired plant. It has worked so far with no dieback noted, except on the targeted weed. You can also find vine/brush killer in a squirt form that works very well for this purpose. It is colored red so you can see exactly where you are applying it.

Use a paintbrush and paint it on the leaves of the plant you want to kill. Has worked for me in the past. You can also segregate the vine of interest by pulling it up through the hole in the bottom of a pot you get from any garden center (old ones that you purchased plants in the past--one gallon sized ones are great). After doing so, paint or spray the leaves of the vine inside the pot. It helps minimize the chance you will get it on the neighboring bush or tree.

Yes, a paintbrush works well! I painted leaves of Virginia Creeper that was creeping where it wasn't welcome in between coral bells. VC is gone and bells are still happy.
I do use gloves though, especially when painting poison ivy!
The flower pot idea is good, but just be darn sure it will never hold a houseplant.
You could also use a disposable aluminum pie plate with a hole cut in it, even shape it a bit more like a funnel to catch any overspray.
BlueBars

Bluebars, I meant those black plastic pots that you get flowers in from the garden centers. I would never use a clay or other type of pot that I would use for potting up a house plant in. Just take paint or even white out and mark that one plastic pot and use it for all instances of when you want to use the Round up or whatever herbicide you want to use!

That's good. You could cut it open from the hole and all along one side to make it even easier to "insert varmint."
BlueBars

As the vine gets older/thicker, does it have thorns? And leaves only on the younger, upper part of the vine? If so, it sounds like cat's claw, which spreads through pink and white tubers. (If you let it grow long enough, it will put out yellow flowers.) The best thing to is to dig up all the tubers, let them dry out, then dip them in RoundUp before disposing of them. If you leave even one smidgen of tuber in the ground, it'll come back, so be diligent!