The weed that you describe is dodder, an annual parasitic plant that derives its nourishment from other plants. It starts from seed each year, but is killed off by frost. Once the dodder has attached itself to the host plant you cannot remove it. (It attached by suction-cuplike suckers). You can cut it out by removing it and the host plant, but because seeds can lie dormant in the soil for several years, the problem likely will return in the future. The best time to control it is before or as it germinates in spring. A pre-emergent herbicide containing DCPA (Dacthal) or trifluraline (Treflan) is fairly effective in preventing germination. (Make certain that the perennials you use the herbicide near are on the label.) Some dodder seeds might escape the herbicide treatment, so keep a close watch for the first signs of the vining stems and remove them before they have a chance to flower and set seed.