Your best bet is to use a systemic weed killer and a plastic bag, cardboard or some other shield for the plants you don't want to kill. Do this at a time when the wind isn't blowing. Make sure you are well protected from the poison ivy by wearing long pants, long sleeves, gloves, etc. Remember that even once the plant is dead, its oils are still viable. When you're removing the dead stuff, cover up again and get everything into a constructor grade trash bag. If any shoots actually come up in the following days or weeks (and they might!) repeat as soon as you see them.
Hi Jon, I would say that if you took a brush and painted the killer on the leaves you would have less chance of killing your other plants.
Cut at the root and apply an herbicide. After the leaves & vines die, you can much more easily remove but they are still able to cause skin irritation so proceed with caution.