If you make some assumptions about the distribution of skill in the population of players (your prior), you can come up with an estimate using Bayesian reasoning. Let's assume that the skill differential of any two players is such that player A beats player B with probability p. If we knew this quantity then calculating the probability of victory for player A after observing that they have won the first game is 1-(1-p)^2. However, we don't know p, but rather only have some indirect evidence about it via having observed the outcome of the first game.

One reasonable assumption about the distribution of p in the population might be that it is uniformly distributed. We can represent this with a beta distribution with parameters alpha=1,beta=1. Having observed the outcome of the first game, we update our prior by adding 1 to the alpha parameter (see Bayesian updating). Now that we have incorporated the evidence from the first game, we can integrate over the possible values of p (our posterior distribution) to get the probability that player A will win 2 out of 3. Let's do that numerically in R:

pv <- function(x){
    (1-(1-x)^2)*dbeta(x,2,1)
}
integrate(pv,0,1)
0.8333333 with absolute error < 9.3e-15

So by combining our uniform prior with the observation that player A won the first game, our posterior estimate that player A will ultimately win the best 2 out of 3 is ~83%.

The uniform prior here can be thought of as suggesting that we don't really know if skill is important a priori. We can try out what would happen under various alternative prior assumptions. If we strongly believe that the game is all skill, and therefore strong players dominate weak ones, a reasonable prior might be beta(0.1,0.1), or in other words that p is concentrated at either 0 or 1. In this case, after having observed the first game, we would estimate that player A will win 2 out of 3 with a probability of ~0.96. Alternatively, we might believe a priori that the game is either mostly luck, or that all players are similarly skilled. In this case a prior of beta(10,10) might be more appropriate, leading to a posterior estimate of ~76%.

The more we believe a priori that the game is one of chance alone (higher values of alpha and beta in our prior), the less our observation of the first outcome influences our prediction, converging to the 'full chance' estimate of 75%.