In the present work, the governing equation of the dynamic response of a buried fluid-conveying pressure pipeline to a transverse earthquake excitation is solved numerically. The structural model of the buried pipe corresponds to the type implemented in hydropower systems. An Euler-Bernoulli beam on elastic Winkler foundation is applied in the transverse vibration model of the buried pipe with appropriate boundary conditions. The constant velocity flow of the inviscid fluid in the pipe is approximated as a plug flow. The Finite Difference Method (FDM) in the form of a fully implicit scheme is applied for the solution of the governing equation of motion. Since this differential equation is of fourth order, two additional mathematical functions are introduced for enabling such numerical treatment. Making use of the implicit FDM with appropriate boundary and initial conditions, the problem converts to a system of algebraic equations with block-tridiagonal structure for each time step within the solution mesh whose right-hand side depends on the results from the previous time step and the earthquake-induced kinematic excitation. The time and space shift of the input seismic excitation over all points of the FD mesh is calculated by means of a special external procedure. For practical application of this computational procedure, a computer program SIVBuPP was written in the MATLAB environment. The numerical algorithm was tested independently by means of a small example and external calculation tools. Further, a numerical example with real structural data and displacement and velocity records from the Duzce 1999 earthquake was solved as implementation of the developed procedure. Finally, conclusions were drawn, and some tasks for future research were formulated as well.