Sounds like bermuda, the bane of my existence. It's what my neighbors planted for a lawn when our subdivision was new, and before I could turn around it had crept under the fences and taken over our whole yard. I tried everything I read and asked on forums and nothing helped. Till I tried newspaper. It brings in a lot of worm activity, which improves the soil. That helps with bermuda because it likes a leaner soil, and also when the soil is looser and aerated the bermuda's easier to pull out.
I spent about two years mulching and pulling and mulching and pulling before I was finallly able to get rid of it, but I did!!
I agree with you about that landscape fabric. It's a waste of money. But you can mulch over the top of it!!
