When we had strawberries growing wild through a garden area I wanted to plant with ornamentals, I pulled as much as I could by hand and threw it out. Then I spread Preen, then mulch on top of that. A few strawberries tried coming back a few times, but after a while they were gone.
Tiffany's idea also sounds good if you can place a plastic tarp (not see through) over the goutweed and smother it. Get rid of all the plant material after that though, so it doesn't pop back up.
