Finite-sum minimization, i.e., problems where the objective may be written as the sum over a collection of instantaneous costs, are ubiquitous in modern machine learning and data science. Efficient numerical techniques for their solution must trade off per-step complexity with the number of steps required for convergence. Incremental Quasi-Newton methods (IQN) achieve a favorable balance of these competing attributes in the sense that their complexity is independent of the sample size, while their convergence rate can be faster than linear. This local superlinear behavior, to date, however, is known only asymptotically. In this work, we put forth a new variant of IQN, specifically of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) type, that incorporates a greedy basis vector selection step, and admits a non-asymptotic explicit local superlinear rate. To the best of our knowledge, this is the first time an explicit superlinear rate has been given for Quasi-Newton methods in the incremental setting.