Very aggressive weeds can penetrate 6" of asphalt and will come through any fabric sold on the market today.
Fortunately, this is a brief and temporary problem, compared with the five to twenty year periods in which you will enjoy unprecedented low maintenance through the use of these landscape materials.
You need only to remove these few weeds when they first appear by pulling them, or by spraying them with Roundup or other post-emergent systemic weed killer. One or two applications should take care of the problem.
Exceptions are grasses such as Bermuda grass, which put out runners and may continually run under, up and through the fabric. These can only be stopped through the use of an edging barrier at least 8'' into the ground.
The use of very light mulch may help the problem, since the aggressive grasses cannot punch through the fabric without top weight to push against.