I would say that you did win using facts. She stopped posting.
If your goal is to merely win an exchange on Twitter by regurgitating information until the other party goes away, then you are right.
If, on the other hand, your goal is to educate the other side through an open and honest exchange, then no, she forfeited, which is not the same as a win.
Our goal has to be to sway them toward at least doing more research rather than taking MSNBC and DailyKOS blogs at face value. She walked away with no acknowledgment that she was wrong, nor she did not say she is now going to look at the pro-gun discussions with a more open mind.
Most Liberals argue with their "sounds logical" information and with snarky comments meant to be sarcastic and demeaning to the other person. I see the same on the other side, but for the most part they are more than patient and only get snarky after the other person starts. Some react positively when they realize you are only giving them a taste of their own medicine, but people like this one will just walk away from the discussion rather than admit you were right from the start.
Disillusionment in a cause like gun control is a hard pill to swallow after they've invested themselves in nothing more than an agenda.