The purpose of the present study is to clarify the fluid flow of an oil damper through numerical analysis in order to obtain an exact value of the damping coefficient of an oil damper. The finite difference method (FDM) was used to solve the governing equation of the fluid flow generated by a moving piston. Time steps evolved according to the fractional step method, and the arbitrary Lagrangian–Eulerian (ALE) method was adopted for the moving boundary. In order to stabilize the computation in the moving boundary problem, a masking method with a single block grid system was used. In other words, algebraic grid generation using a stretching function was used for the moving piston in the cylinder of the oil damper. The time-dependent coordinate system in the physical domain, which coincides with the contour of the moving boundary, is transformed into a fixed rectangular coordinate system in the computational domain. The computational results were compared with experimentally obtained results and the approximate analytical solution. The results of the present analysis exhibit good agreement with the experimental results over various widths of the annular flow channel between the piston and cylinder.