This paper is concerned with the finite-time stabilization issue of stochastic coupled systems on networks with Markovian switching via feedback control. The aim of this paper is to design a state feedback controller to stabilize the states of such stochastic coupled systems on networks within finite time. Focusing on the finite-time stabilization issue, this paper utilizes Kirchhoff's Matrix Tree Theorem and Lyapunov method to establish two sufficient criteria. Based on these criteria, the relationship between the time to reach finite-time stabilization and the topology structure of the network can be shown. Furthermore, to verify our theoretical results, an application to a concrete finite-time stabilization problem of stochastic coupled oscillators with Markovian switching is presented. Finally, a numerical example is given to illustrate the effectiveness and feasibility of the proposed results.