When approaching to problems in computer science, we often encounter situations where a subset of a finite set maximizing some utility function needs to be selected. Some of such utility functions are known to be approximately submodular. For the problem of maximizing an approximately submodular function (ASFM problem), a greedy algorithm quickly finds good feasible solutions for many instances while guaranteeing ($1-e^{-γ}$)-approximation ratio for a given submodular ratio $γ$. However, we still encounter its applications that ask more accurate or exactly optimal solutions within a reasonable computation time. In this paper, we present an efficient branch-and-cut algorithm for the non-decreasing ASFM problem based on its binary integer programming (BIP) formulation with an exponential number of constraints. To this end, we first derive a BIP formulation of the ASFM problem and then, develop an improved constraint generation algorithm that starts from a reduced BIP problem with a small subset of constraints and repeats solving the reduced BIP problem while adding a promising set of constraints at each iteration. Moreover, we incorporate it into a branch-and-cut algorithm to attain good upper bounds while solving a smaller number of nodes of a search tree. The computational results for three types of well-known benchmark instances show that our algorithm performs better than the conventional exact algorithms.