The recently proposed first-order mean spherical approximation (FMSA) [Y. Tang, J. Chem. Phys. 121, 10605 (2004)] for inhomogeneous fluids is extended to study the phase behavior of nanoconfined Lennard-Jones fluids, which is consistent with the phase equilibria calculation of the corresponding bulk fluid. With a combination of fundamental measure theory, FMSA provides Helmholtz free energy and direct correlation function to formulate density functional theory, which implementation is as easy as the mean-field theory. Following previous success in predicting density profiles inside slit pores, this work is focused specially on the vapor-liquid equilibrium of the Lennard-Jones fluids inside these pores. It is found that outside the critical region FMSA predicts well the equilibrium diagram of slit pores with the sizes of 5.0, 7.5, and 10 molecular diameters by comparing with available computer simulation data. As a quantitative method, FMSA can be treated as an extension from its bulk calculation, while the mean-field theory is only qualitative, as its bulk version.