A molecular simulation methodology with the algorithm usually used in equation of state calculations for phase equilibrium is applied to calculations for vapor-liquid equilibria of Lennard-Jones binary mixtures. This method can determine a phase equilibrium of pure substances or multicomponent mixtures by performing a single simulation, and any kinds of methods to determine chemical potentials can be used. The calculations for vapor-liquid equilibria are performed by simulation, with results in agreement with those by a Gibbs ensemble method except near critical points for the mixtures. Good reproducibility of the simulation results by the proposed method is obtained.