This article aims to enlarge the applicability of event-triggered control for stochastic systems, and is particularly concerned with event-triggered stabilization for stochastic systems with control-dependent diffusion term. Roughly treating stochastic noises as unfavorable factors for system stability, the related works are based on Lyapunov-type conditions for ISS property with respect to sampling error as those in the nonstochastic context. The conditions imply the existence of a twice-differentiable Lyapunov function with negative-definite infinitesimal under continuous-time control, and indeed exclude many familiar stochastic systems with control-dependent diffusion term which are continuously stabilizable while no such Lyapunov function exists. As the main contribution of this article, an enlarged framework of event-triggered stabilization is established for stochastic systems with control-dependent diffusion term, starting from the system stability under continuous-time control, rather than from Lyapunov-type conditions for ISS property. Specifically, with moment exponential stability assumed under continuous-time control, elementary event-triggering mechanisms are proposed with an enforced minimum inter-execution interval, under which the event-triggered controller can render the system exponentially stable both in the sense of expectation and in the almost sure sense. Particularly, a distinctive analysis of closed-loop stability is performed via delicate comparison between the solutions under the event-triggered, and continuous-time controls. Within the framework, inclusive Lyapunov-type conditions incorporating the underlying positive effects of stochastic noises are presented for event-triggered exponential stabilization of the stochastic systems, where the candidate Lyapunov function involved is weakened to be nondifferentiable at the origin and, even, to allow non-negative infinitesimal under continuous-time control.