It's not a wonderfully detailed photo but it looks a lot like bindweed, Convolvulus arvensis. And yes, that is a very difficult weed to control, primarily due to a deep and far reaching root system.

Bindweed is a horror.That unhealthy alien-looking root is part of a larger root that sends up the shoots and leaves every so often. People swear by gathering up all the top growth and soaking it in Round-Up, and I have had some little success with a drop of straight RU on a cut shoot, but the only thing that gets it under control is digging up the whole area and pulling out the root - gently, because it will break and regrow from any little bit. Unfortunately, unless my neighbor does the same digging up of the root, which he won't, I will keep dealing with incursions, but at least I've got it narrowed down to one area, rather than the whole width of the back fence.

If you haven't planted yet, maybe you can do the Big Dig this spring; if not, just keep the little creepy vines pulled up and start to exhaust the root that way. I wish you all the luck in the world in dealing with this.

Pulling won't fix it. If you don't want to use herbicides you are left with digging. Use a fork and keep your eyes peeled for any fraction of root. Don't leave even a half inch.