I don't really know what's the width of the "cracks" you're talking about, but if it's really the narrow cracks I'm assuming it is, where the two pieces of concrete are pretty much still abutting each other, you're going to have difficulty cultivating anything there. The seeds of weeds may blow into small holes and grass shoots may find their way up, but it's not going to be easy for any gardener to try and "plant" anything in that space. You might be able to find a particularly aggressive plant that would be able to grow roots beneath the pavement and come up through the cracks, but I'd worry that anything that would be able to do that might be too aggressive for the rest of your garden.
If you want to stop fighting the weeds, I'd advise tackling the pavement itself. Either fill in the cracks, or widen them out so that you can really grow something there (like growing ground cover between paving stones). Filling in the cracks isn't that difficult (here is a pretty good instructional video I found on YouTube after a quick googling) and should hold up pretty well for a few seasons. Widening out the cracks would take a bit more mechanical work
In general, I've dealt with weeds and grass coming up in pavement cracks with an electric weed whacker and a hori-hori knife. I've also used a wire brush on a handle like this, but it does take a bit of elbow grease to wipe out weeds. You can also use a propane torch, and of course, there's always herbicides, but I tend to stay away from those as much as I can.
Update (based on comments): my answer does show the hole in my knowledge, that I don't know what plant might make a good choice for your particular climate. I'm sure there are several that will work in this climate and can grow in those gaps. I'm particular to dwarf mondo grasses and creeping thyme varieties, both of which can grow into narrow spaces.
That said, I doubt that you'll find anything that will completely crowd out the weeds, and something that's very aggressive might bring on other issues. I've fought my own battles with unwanted plants coming up in the middle of ground covers, and have found that nothing is going to completely replace some tending to pull out weeds or other invaders. They will help keep out some of the weeds, and will certainly look better than what you have now, but don't toss out that patio crack weeder yet.