Remove the vine manually. Some vines are easy to dig out, but if a piece is left behind, it will resprout. Many vines also have tangled root systems or stems that root every few feet and create new plants. Mechanical control in this manner is very sporadic.
Go over the area with a weed trimmer or mower before flowers have turned to seed. Rake up the debris and dispose of it to prevent rooting of vegetative pieces. In two weeks, come back and mow or trim again. Do this consistently over the growing season to minimize the vigor of the root system. Annual vines are easier to get rid of this way, but perennials come back from the root, so it may require another season of mowing to sap the roots of energy.
Use a two-part method to be sure you have gotten rid of the nuisance vine. Cut the vine to within 12 to 16 inches of its base. Use gloves and wear long sleeves and long pants in cases where the vine or sap are toxic, such as poison ivy. Destroy the severed vine.
Immediately apply a mixture of glyphosate and water. This herbicide is nonspecific, so care must be taken to spray only the open wound of the plant. Use a paintbrush if other plants are nearby, and paint the solution directly on the open wound.
Spray the leaves of plants that are not near desirable plants. For example, a field of vines that has nothing else planted in it can have this type of foliar application. Use a backpack sprayer loaded with the same glyphosate solution. Reapply the herbicide in two weeks, and thereafter as needed.
Apply cultural control in the forms of withholding water or pruning to let light on shade vines. Reverse the conditions that the vine finds so habitable. This is rarely easy to do, because it adversely affects nearby plants, but it may work in some situations.