You may be able to get completely rid of that by letting the long strands "drink" poison from a jar.

Here's a well-written quote, "glass jar with a good, not concentrated shrub killer in it. Slip the
branch into the jar. The thought is to wick the toxin into the roots
directly. I found that concentrate kills the branch before it 'wicks.'

Doing this, one finds out which bindweeds are attached to which and how many sub-surface individual networks there are..."
- http://forums.gardenweb.com/discussions/2131246/whats-the-best-way-to-get-rid-of-bindweed

This method should work with any of the glyphosate products, that
function by being absorbed systemically by actively growing leaves.
This allows only the exact amount needed to be used, and not applying
anything to anything else or the ground. You would dig a little divot
for the jar, so it doesn't tip over, then cover the whole thing (gently,
so you don't injure those vigorous branches that need to drink a lot of
gunk,) with foil or a plastic bag, to keep rain out.

You may need to repeat this several times during the growing season,
and possibly into next year, depending entirely on its' size, vigor, but
there are many testimonials that it does work if done well &
diligently.

In the meantime, cut off any flowers, to prevent them from making new
seeds that you'll have to battle in the future.

There is no reason why you cannot employ triclopyr (which is available under various brand names) as a paint-on herbicide regardless of its formulation. AMOF, the concentrates used in this manner (just dipping a sponge tipped paintbrush in the full strength product and painting cut stems) should be even more effective.