Cut the Asian jasmine down as far as possible. Asian jasmine grows quickly and can quickly overtake other plants or even a house. Killing it requires cutting it down as far as possible so that it can be more easily removed without damage to other plants or a house. A lawnmower on a low setting will usually work, though the Asian jasmine can be cut down with other tools like a weed whacker.
Use a tiller to break through the soil. The roots will be well settled into the soil, so it can be a little difficult to get started. Once started, tilling should become easier. This makes it easier for removal.
Pull the roots out. The only way to kill the Asian jasmine is to pull out the roots. Plant killers are often ineffective, but pulling the plant out by the roots will usually get rid of it . Tilling beforehand makes removal of the root much easier.
Raise the pH balance of the soil. Asian jasmine is not picky about soil; but it struggles in soils with pH higher than 8. If you raise the pH high enough, you can kill the plant off completely. Then you can rebalance the pH for other plants after a season.