Here is a short guide about how to do it better:
http://www.improveyoursocialskills.com/conversation/conversation-flow
I'm not sure how I've gotten away with not having internalized all this stuff for so long.
Addendum
I later got some very good advice about this -- focus on the situation. A useful technique for getting over the anxiety of a stilted conversation is to express an opinion of something nearby ("I hate that building, it's oppressive" or "these chairs are comfortable, I like them") or something that is happening. Getting over that first-sentence hump was a big deal for me -- I would get anxious about not talking, and the feeling of almost panic made it even harder to think of something to say. Focusing on the local environment or current situation and picking something out to discuss helped a lot.
Sharing an opinion also effectively invites the other person to share their own opinion.