Here are several solutions:
1. Spray a brush killer on it where you can.
2. Take a small paint brush and paint it full strength on the leaves (less collateral damage, but drips can kill other plants if they land on leaves).
3. Dig it up (wear protection - no gaps in clothing and be careful taking clothing off!). You have to dig up the underground stems, or they'll send up shoots next year. They're only an inch or two below the surface, but they're rather too easy to break.
4. Cut vines growing up trees near the base and apply full strength brush killer to the cut. (Do this even if the tree is out of the way. It will reduce the number of seedlings you get each year.)