Thank you for your question and pictures. 

We cannot see one vine in the pictures you have transmitted. Please pluck the vine out of the ground. Place it on a white piece of paper and retake a few photos so we can identify the plant in question.   

Kerry,

I took additional pictures as you suggested and attached them to my response.

/Ed

Thank you for the additional pictures.

There is very little foliage this time of year, and obviously not in bloom.
A colleague sent me the following note and I agree.

Note: Kerry, my guess is that this "vine" may be the stolons of Whiplash Daisy aka Creeping Fleabane (Erigeron flagellaris). It looks like stolons are rooting where they touch soil, to form new plants. Whiplash daisy is listed as occurring in El Paso County.

Here is a link with pictures from Western University of New Mexico
http://wnmu.edu/academic/nspages/gilaflora/erigeron_flagellaris.htmls

Complete control is difficult :hand removal may be the easiest and most economical way to control it. When few plants are present or if you see a potential new weed, dig it, pull it, or remove the seedhead before the seed can disperse. This technique works particularly well for annuals (creeping fleabane is an annual) . Properly dispose of weeds after removal to prevent seed or vegetative structure dispersal. Do not compost pulled weeds. Put them in a bag and close the bag so the seeds don't get out.

Key points about mowing and hand removal:

• Repeated mowing reduces competitive ability, depletes root carbohydrates, and prevents seed production.
• If you see a new weed, dig it, pull it, or remove the seedhead before seeds can disperse.
  

Herbicides

Herbicides provide a convenient, economical, and effective way to help manage weeds.Herbicides may not be a necessity but without the use of chemical weed control, mechanical and cultural control methods become that much more important. 

Controlling the problem. Current herbicide registrations for control of fleabane are limited to Touchdown Hi Tech (glyphosate 500g/L). Spray.Seed and paraquat (eg Gramoxone) labels are worded such that they are registered for most annual weeds. This would include fleabane as it occasionally is referred as an annual weed in response to dry seasons.