Ask Mr. Smarty Plants

From only your description I can't identify your vine and whether it is native or introduced.  If you want us to identify it, please visit Mr. Smarty Plants' Plant Identification page to read instructions for submitting photos.  With photos of your vine, we will do our very best to identify it. 

It hardly matters, however, what the vine is since it is being very invasive and, from your description, it doesn't sound as if it's going to be an easy job to get rid of it.  You obviously can't spray it with herbicide unless you want to lose your shrubs as well.  This means you are going to have to deal with it piece by piece.  The best thing you can do is to be vigilant and keep the vine from producing fruit by removing it before it flowers and sets the fruit.  This will keep you from spreading the seeds when you pull it out of the shrubs.  This will require pulling it out of the shrub early and, if possible, digging it up by the roots.  Simply pulling out the roots is probably not going to work since you say it breaks easily so you're going to have to dig.  If you have to remove vines with the fruit still on them, cut and gather the fruit BEFORE you pull out the vines.  Put the berries into a plastic bag, seal it and put the bag in your trash.  

An alternative to digging the roots of the vines out of the ground is to cut them at or near ground level and paint the surface of the cut with a 15-25% solution of glyphosate herbicide (available at most plant nurseries—one brand name is Roundup).  This allows you to target the vine without (if you are very careful) getting the herbicide on your shrubbery or other desirable plants.  This works best in summer and fall for deciduous plants and year round for evergreen ones. We urge you to read and follow safety/environmental instructions on all herbicide labels.

North Carolina Botanical Garden has an excellent booklet, Controlling Invasive Plants, with descriptions and photos of invasive non-native plants common in North Carolina and methods to control them.  This booklet is available on-line in PDF format.

Vigilance and persistence are the keys to getting rid of this pest.  Best of luck!