"Strangle Vine" (Dodder) on Herbs

I live in northern Virginia and have an herb garden with a variety of things growing quite well. However, the last two years I have had a problem with a thin, fast-growing yellow vine that wraps itself around the leaves of chive or the stems of oregano or lavender (for example) and grows very rapidly. It looks like a very thin piece of string with few to no leaves, but will occasionally have very very small white flowers on it or clusters of what appear to be very small seeds. If I don't keep on top of it, it will spread very rapidly through the chives, oregano, lavender, etc.
It will wrap around the stem from one to four times and then head off to another stem, where it will do the same thing. My wife and I have called it "strangle vine" due to the nature of its growth, but don't know what it is or how to prevent it. I will occasionally try to follow the vine around to see where it is coming out of the ground, but have never found where it starts.
Do you have any idea what this is and how to combat it?

The "strangle vine" sounds like one of the dodders (Cuscuta spp.) that are known to parasitize plants. Dodder causes serious damage to crops world wide.
It is spread by birds and by seeds. After the seeds germinate the seedlings climb up nearby plants and form knobs on the host plants where they will penetrate. Once the seedlings have penetrated into the hosts their roots fall away and the plants become fully parasitic. They will produce many strands of orange or yellow stringlike growth wrapped around the host plants. Small clusters of white flowers, as you have observed already, produce seeds which go on to produce more parasite plants.
Sometimes if dodder is noticed early it is possible to eradicate it by cutting out the strands before they produce seeds. But in heavy infestations the only option is to remove and burn the infected plants. It is so despised that if a single dodder seed is found in commercial seeds the entire lot of seeds is prohibited from sale and has to be destroyed.