Oh wow, that sounds a bit like zoysia grass. It creeps and forms a mat. So this is good then for your particular climate? The website says it grows well in sandy soil. Is that the type of soil you have? We have heavy clay soil here, cold winters, and zoysia really isn’t a good choice here. It will go dormant and turn brown for the winter and people don’t like its slow return in the spring (other lawns will be green and the zoysia will still be brown). That being said, it is very drought resistant. So I suppose really the issue is an aesthetic one (it survives and needs little care, but it doesn’t always look good). In other areas of the country, where the winters aren’t so cold, the grass actually looks pretty nice year-round.
Okay, so I can tell you what people do here for fescue lawns (probably the better choice since it’s too hot and dry for bluegrass and natives like buffalo grass don’t hold up well to foot traffic). In late summer, you start doing weed control. This means hand-pulling of weed scattered sporadically. If there is a large patch of pretty much nothing but weeds you can use horticultural vinegar or horticultural soap. You have to be careful though because they are non-selective herbicides and will kill grass. You can also use boiling water. Again, it will kill the grass too so watch where you’re pouring it.
Next you want to aerate the lawn in early fall. You can do this because most weeds are done doing their seeds so there are less around to try to take over. Since we have clay soil here, it’s recommended that you spread a thin layer of compost over the lawn and then overseed it. When done in the early fall, it gives cool season grasses like fescue a chance to fill in rapidly before the growing season ends (i.e. winter comes). The idea is to give the grass a chance to fill in, leaving less room for weeds.
Then in the spring, you can do corn gluten meal as a pre-emergent. You have to do this very early, before weed seeds that may be present have a chance to germinate. The corn gluten meal is also a fertilizer, so don’t fertilize in the spring. This just causes excessive growth. Allow the grass to be taller by maintaining a mowing height of at least 3 inches. The taller the grass, the more shade down by the soil. You also want to use a mulching mower because the grass clippings being returned to the lawn act as fertilizer and they help block out the weeds too.
Anyway, this is what you do for lawns where I am. It can often take several years to get a big weed problem under control this way, but people do have success. It just takes a certain level of dedication.
I don’t know if this is helpful at all since I’m not familiar with the grass you have or what the conditions are like there.