Thank you for your question about the weed you have. I apologize that it has gone unanswered for 17 days. It has bounced around in the system for a while, and then it was assigned to me while I was out of town.
We will need a little more information to I.D. this weed and tell you how to treat it. Weeds in yards and gardens and fields all have different considerations when you want to eliminate it.
Also, although your photos were excellent for showing roots and structure, I could not see the leaves as they had wilted by the time you took the photo. If you could take a photo of the weed in place, that would be helpful.
In the meantime, the university of Minnesota has a good weed identifier for our general area. http://www.extension.umn.edu/gardeninfo/weedid/
Here is a link to a general article on weed control in lawn and garden http://www.sdstate.edu/ps/extension/weed-mgmt/lawn-garden-weed-mgmt.cfm
I look forward to hearing from you,