I had the same problem, and boy, was I surprised! No listing ever mentioned runners. Anyhow, I cut down the main, first vine, and all the runners, then hit each stump with a drop or two of straight Round-Up concentrate - not one came back up. If you miss one or two (I panicked immediately, and got rid of it after the first crop of runners), I would think that hitting them with RU as soon as you see them would work.

Going off the recommendation from Ken_adrian, a frequent poster on Gardenweb, I used an old squeeze bottle, like a dish detergent bottle, to apply the RU, so as not to get it all over everything, just on the stumps.

Sounds like a plan. I am afraid it is going to consume my garage & flower beds if I don't do something drastic.

It probably will!