I did some quick research. Here is a good article on the problem, and gives a number of good suggestions on different types of control.
http://aquat1.ifas.ufl.edu/node/405
Apparently, this stuff re-germinates from almost any part of a root, so burning them out is one the suggestions for control. Since that is not probably an option in your area, you should consider using a lawn product for broadleaf weed control that contains triclopyr. There are a number of products that will do the job nicely.
Woodweasel has convinced me that I should recommend covering an area with black plastic sheeting: weight it down, leave it there for a week or more, and that should do in the offending plants that were covered. You will, of course, need to re-seed the area.
If seedlings are the only problem, you should consider carefully what kind of pre-emergent herbicide will reduce the threat of future sprouts once you have the seedling eliminated.
I'm not from that part of the world, so I would be speculating on what would work best.
Bermie: according to the article, even small chunks of root will re-grow, so mowing at -1" would probably cause them to proliferate! It would be fun to watch though. I wonder if that would look like one of those fancy concrete finishing machines zooming around the lawn?