This paper studies integral input-to-state stability of stochastic nonlinear systems in the Stratonovich sense and develops Lyapunov-type criteria. Instead of merely extending previous results on deterministic systems and stochastic systems of Itô type, this paper focuses on the unique feature that allows one to stabilize a system globally by stochastic noise when the system is not stable globally in the absence of the noise. Making use of the Lyapunov-type characterization of integral input-to-state stability developed in this paper, the capability of admitting unstable subsystems in establishing stability of interconnected systems is also pursued. Numerical examples are presented to illustrate these curious points.