In this paper, an analytical method for evaluation of via-plate capacitance including finite plate thickness in multilayer structures is proposed. An electrostatic boundary-value problem associated with a via in parallel plates is solved based on the Hankel transform and mode matching method, and thereby a fast-converging series-form expression for the via-plate capacitance is derived. The effect of finite plate thickness is taken into account by adopting the mode matching method instead of imposing an artificial perfect magnetic conductor boundary as in the previous works. We perform some computations to show the behaviors of the capacitance in terms of the via geometry, and they are validated through comparison with a static/quasi-static simulation, full-wave electromagnetic simulation, and the previous analytical solutions of via capacitance.