Use pruners, loppers or a handsaw to cut off the wisteria vine. Leave 2 inches of stump when you cut the vine.
Use a shovel to dig out smaller stumps and roots from the ground. The more you can manually remove, the quicker the remaining roots will die from lack of food and sunlight.
Apply a glyphosate herbicide to the cut surface of the stump. The ratio of herbicide to water is one part herbicide to three parts water. Use a sprayer, or brush the herbicide on with a paintbrush. Glyphosate herbicides are absorbed and carried throughout the plant, eventually killing the roots.
Repeat this application on the stump if new growth appears while the glyphosate is taking effect. Along with coating the stump surface, apply the herbicide to the new growth.
Hand pull or spray with the glyphosate herbicide any new sprouts from the roots. Eliminating all foliage will starve the roots, and they will die.