Unravel the vine from any plants you want to keep. Use pruning shears to cut the vine as needed, but don't pull the vine off the desirable plants, as doing so may damage the plants you want to keep. Be sure to wear gardening gloves to protect your hands from the sharp thorns.
Spray or paint the leaves and stems with a non-selective herbicide. Be sure to protect the plants you wish to keep from any overspray, as non-selective herbicides kill any vegetation they contact.
Allow the herbicide to dry on the plant for one to two days.
Fill floral water tubes with undiluted herbicide. Wear gloves and eye protection to protect yourself from any splashes.
Cut back the Smilax stems, leaving only 8 to 10 inches above the ground.
Peel the bark from the stems and cut each stem at a 45-degree angle.
Place each stem into one of the herbicide-filled floral picks. You can leave them hanging on the plant or drive the pointed end of the water tubes into the ground to anchor them. Leave them in place until the plant is clearly dead or the herbicide disappears from the tubes.
Carefully dig or pull up the dead plants, removing as much of the root system and attached tubers as possible.
Repeat as needed. It may take several tries to completely eliminate Smilax vines from your garden.