The paper addresses a continuous-time continuous-space chance-constrained stochastic optimal control (SOC) problem where the probability of failure to satisfy given state constraints is explicitly bounded. We leverage the notion of exit time from continuous-time stochastic calculus to formulate a chance-constrained SOC problem. Without any conservative approximation, the chance constraint is transformed into an expectation of an indicator function which can be incorporated into the cost function by considering a dual formulation. We then express the dual function in terms of the solution to a Hamilton-Jacobi-Bellman partial differential equation parameterized by the dual variable. Under a certain assumption on the system dynamics and cost function, it is shown that a strong duality holds between the primal chance-constrained problem and its dual. The Path integral approach is utilized to numerically solve the dual problem via gradient ascent using open-loop samples of system trajectories. We present simulation studies on chance-constrained motion planning for spatial navigation of mobile robots and the solution of the path integral approach is compared with that of the finite difference method.