If we miss an underground runner of our Wisteria and it raises its Hydra head in our juniper border, I seek it out and deal with it like this: I follow the main stem back toward the mother plant and dig it up. I then dig a small depression around it. Using pruning shears, I remove a section of the runner/stem. This keeps the desired plant from being affected. I then place the targeted plant's end in a container and fill the container with Roundup or another suitable herbicide, adding more liquid until it stops absorbing it, usually 2-3 days. Within a week the target plant withers and dies off.
A variation of this method has been successful in eliminating small infestations of Ailanthus, an invasive with a suckering habit. Holes are drilled into the suckers and tightly fitted sections of straws are inserted into the holes and filled with herbicide. As the herbicide is absorbed, keep refilling the straws until the plant becomes inactive.
As in all situations with chemical agents, care is required: Use personal protective gear and prevent accidental contact with pets and children. An overturned crate or an overturned wheelbarrow, in case of rain, can protect the worksite and prevent the contamination of the area. After the herbicide is mixed to the recommended concentration, it is relatively safe, and applications such as these give you a lot of control on treating only the target plants. After the job is done, be sure to follow recommended procedures in the disposal of the materials.