How can I stop invasion of zoysia grass?
How to Kill Zoysia Grass
Certified GKH Gardening Expert
Zoysia grass is a very invasive grass. It can be difficult to control. Small areas of Zoysia can be treated with a non-selective type herbicide, such as Round-up. However, this type of herbicide can potentially kill any plant it comes in contact with, so be careful when spraying around plants you want to keep!
For best results, apply the herbicide when the Zoysia grass is still green, actively growing and not dormant or under drought stress. Since Zoysia grass may regrow, more than one application will probably be needed. Since Round-up dissipates rather quickly, it shouldn't pose any problems with future planting. This article may also help: https://www.gardeningknowhow.com/lawn-care/specific/zoysia/controlling-zoysia-grass.htm
Larger lawn areas can be controlled by putting a tarp over it at any time, even after the grass has started to come out of dormancy. The lack of light will kill it.