The properties of random graphs provide insight into the behavior of real-world complex networks. One such property is the Typical Distance, which characterizes the time required to traverse the network. For example, the Typical Distance measures how fast a virus spreads through a population. The probability that the Typical Distance is large is difficult to estimate via crude Monte Carlo. We propose a new sequential importance sampling estimator that can find the probability of a large Typical Distance in preferential attachment models, with a computational complexity that is quadratic in the number of nodes. Numerical experiments indicate that the estimator is significantly more efficient than crude Monte Carlo.