We investigate stochastic averaging theory in continuous time for locally Lipschitz systems and the applications of this theory to stability analysis of stochastic extremum seeking algorithms. First, we establish a general stochastic averaging principle and some related stability theorems for a class of continuous-time nonlinear systems with stochastic perturbations and remove or weaken several significant restrictions present in existing results: global Lipschitzness of the nonlinear vector field, equilibrium preservation under the stochastic perturbation, global exponential stability of the average system, and compactness of the state space of the perturbation process. Then, we propose a continuous-time extremum seeking algorithm with stochastic excitation signals instead of deterministic periodic signals. We analyze the stability of stochastic extremum seeking for static maps and for general nonlinear dynamic systems.