Posted: 17/04/2015 at 12:20
I would have thought that weedkiller is your best option unless (as Pansy has already said) the area is sloping so the liquid just runs off.
Assuming the area is reasonably level - I would grit my teeth and let the weeds come through for a few weeks so there is a bit of growth. I would then apply Path Clear - this has a residual effect & helps prevent seeds germinating. Use the concentrate (rather than ready mixed) & apply using a large sprayer - you can cover a large area fairly quickly. Choose a day when it's a bit overcast but no rain is forecast (& no wind of course) - I often do mine in the evening.
If you use Path Clear you can spray all the grout joints (even those that appear 'clean') as well as the weeds to try & stop seed germination. The weeds should start to die within a few days. I would then leave it a few weeks and respray if weeds start to germinate. After that I would check the paving and regrout any areas where there are holes and gaps for weeds to set up home.
Glyphosphate / systemic weedkillers are excellent for killing weeds in ground that will be cultivated but only works on plants with green foliage. It doesn't stop seeds germinating & it becomes inactive when it comes into contact with woody stems or the soil. I would only use that on the persistent weeds you think are growing from under the paving itself once you've tried the above.
I find I need to apply Path Clear about 3 times between early spring and late winter on my gravel drive (aka a "weed seed tray").