Depends if it's someone you're feeling comfortable with or not.
I don't think you'd be self conscious of the awkward silence if you were comfortable with the other person (I probably won't), but if you do you can always joke about it.
If you're not comfortable with the other person (yet), I usually think of a few topics to present before I go to fall back on, but that's just preventing it and not reacting on it, I don't really know how to react on it myself.