A rigorous formalism is presented for sensitivity analysis of functional-type responses associated with the well-posed system of quasi-linear partial differential equations (PDEs) of hyperbolic type that describe one-dimensional, two-phase flows. The rigor and generality of this formalism stem from the use of G differentials. In particular, it is possible to treat discontinuities and parameters that are functions rather than scalars. This formalism uses adjoint functions to determine efficiently sensitivities to many parameter variations. The adjoint system satisfied by these adjoint functions is explicitly determined and shown to be solvable as a well-posed system of linear first-order PDEs possessing the same characteristics as the original quasi-linear PDEs. For completeness, a general solution of this adjoint system is obtained by using the method of characteristics. The physical meaning of this sensitivity analysis formalism is illustrated by an application to the homogeneous equilibrium model for two-phase flow. Although this formalism does not address transition phenomena between single- and two-phase flow regimes and ignores higher order effects of parameter variations, it provides a complete theoretical framework for implementing an efficient sensitivity analysis capability into one-dimensional, two-phase flow models.