Negative binomial wouldn't work. I don't know of a 'nice' way to do this. There are a few options though but all require some sort of enumeration of what could possibly happen.

My initial thoughts were: 1) just to enumerate all the possibilities 2) calculate the probability of each possibility and then 3) sum the probabilities of each possibility that meets the criteria. This is easy enough to do using a programming language.

I then thought that one could use the fact that "first to six" is the same as "best of 11". So if we can just figure out the probabilities of A winning 0, 1, 2,... 11 in a set of 11 games then sum the probabilities for >= 6 we will get our answer. We can use excel and some relatively simple formula manipulation to do this relatively quickly by recognizing that in round X the probability that A has won Y games is: P(A won Y-1 games in round X-1)*P(A wins this round) + P(A win Y games in round X-1)*P(A loses this round).

I made a spreadsheet in excel to do just that. You can control the probabilities that "A" wins each round so you could even play around with having an entirely different probability for each round instead of just flip-flopping probabilities between rounds. For your example the answer that A wins comes out to be 0.630551086 but I suspect you don't care about that particular example and are more interested in how to get the answer so you could play around with what happens for different probabilities.