The problem with goats is that they eat everything. I borrowed a couple a few years back to clear blackberries and they did the job, but nothing under 5 feet was left alive. And the blackberries grew back!
The only way I know of is to suffocate it by putting cardboard weighted with rocks over it for a few weeks, but that on a big area that's a lot of cardboard and a long time to go without a lawn. And if you don't get it all the problem can easily return.