We are interested in multiphase flows with liquid-vapor transition, such as cavitating, flashing and boiling flows. We describe these flows by a hyperbolic single-velocity six-equation two-phase compressible flow model, which is composed of the phasic mass and total energy equations, one volume fraction equation, and the mixture momentum equation. The model contains pressure, temperature, and chemical potential relaxation source terms accounting for volume, heat and mass transfer, respectively. The model equations are numerically solved via a fractional step algorithm, where we alternate between the solution of the homogeneous hyperbolic portion of the system via a HLLC/Suliciu-type finite volume wave propagation scheme, and the solution of a sequence of systems of ordinary differential equations for the relaxation source terms driving the flow toward mechanical, thermal and chemical equilibrium. We design numerical procedures for the relaxation terms based on semi-exact exponential solutions, which can efficiently describe arbitrary-rate heat and mass transfer, both slow finite-rate processes and stiff instantaneous ones. For instantaneous processes we show the capability of the numerical model to approximate efficiently solutions to the relaxed two-phase flow models that can be established theoretically from the parent six-equation model in the limit of instantaneous equilibria. Several numerical tests are presented to show the effectiveness of the proposed numerical method, including simulations of depressurizations leading to metastable superheated liquid, and a two-dimensional simulation of a fuel injector.