Modeling the transient response of compressible fluid systems using dynamic systems theory is relevant to various engineering fields, such as gas pipelines, compressors, or internal combustion engines. Many applications, for instance, real-time simulation tools, system optimization, estimation and control would greatly benefit from the availability of predictive models with high fidelity and low calibration requirements. This paper presents a novel approach for the solution of the nonlinear partial differential equations (PDEs) describing unsteady flows in compressible fluid systems. A systematic methodology is developed to operate model-order reduction of distributed-parameter systems described by hyperbolic PDEs. The result is a low-order dynamic system, in the form of ordinary differential equations (ODEs), which enables one to apply feedback control or observer design techniques. The paper combines an integral representation of the conservation laws with a projection based onto a set of eigenfunctions, which capture and solve the spatially dependent nature of the system separately from its time evolution. The resulting model, being directly derived from the conservation laws, leads to high prediction accuracy and virtually no calibration requirements. The methodology is demonstrated in this paper with reference to classic linear and nonlinear problems for compressible fluids, and validated against analytical solutions.