Round up won't work I'm afraid - it might check the topgrowth, but it won't kill the roots, and yes, any roots present will grow. Round up does not persist in the soil, (though there's some argument about that currently) but any which gets onto leaves of other plants will kill those too. It's not clear to me from your question whether or not you're able to get good access to the main roots at the base of the plant - if you can, then some heavy duty, persistent and rigorous digging out is called for. If you find a main, very woody large root, that can be treated with a different product to kill it off, but this plant usually spreads through the ground, with roots covering a large area.

Try what Bamboo has said: if that fails you could try and suffocate the Russian Vine.

Get some newspapers and thickly cover the area around the vine and soak it with a hose. When you have a mushy surface lay some cardboard on top of that and soak again. Leave it re-soaking it as it dries for 4-6 weeks (you may need to hose it 2-3 daily). This will suffocate everything under the newspaper/cardboard. You can work it into the soil or remove it, removing any roots not rotted.