We have the same problem in our hay field, this weed is actually poisonous to animals and to eradicate it , we sought advice from are local Agricultural supplers. Most weed killers have little effect, the one they reccommended is Pastoral, a double strength dose in late spring leaves the pasture clean enough to make hay. But as Thunderbird says sometimes more than application may be neccessary.
As this plant has underground runners, you may clear it from one area only to have it return in another. You may never fully eradicate it.