The 14 partial differential equation system describing the longitudinal–flexural–torsional dynamic behavior of liquid-filled pipelines contains coupled equations due to mutual boundary conditions and Poisson contraction ratio terms. Solutions of the above system are available in the frequency-domain or in the time-domain (method of characteristics (MOC)). In this paper, an analytic solution in the domain of time and space is achieved. Double integral transform, namely, finite sine Fourier transform (FSFT) and Laplace transform, is applied to the derived system of the 14 fourth-order partial differential equations, yielding an algebraic system in terms of the transformed variables. The inversion of the FSFT is an easy task, but the analytic inversion of the Laplace transforms is very challenging. Both integral transform inversions of the 14 transformed variables are successfully performed, and an analytic matrix formula in the domain of time and space along with numerical results is obtained.