If you could kill the weeds with some sort of spray - Round up is good, but don't let it drift to something you don't want to kill - but killing before teh weednetting is the best thing; then nothing should "grow through"... but use the weednetting, newspaper - mulch and plant through the holes; works great. and looks so neat.
You can catch overspray if you have a big box, bottom removed, you could set over an area to spray - flip the lid shut and leave it for a while - continue on down the area you want "dead" doing this over and over - keeps the spray from drifting. Allow a couple days before continuing with project.