We consider a kind of stochastic exit time optimal control problem in which the cost functional is defined through a nonlinear backward stochastic differential equation. We study the regularity of the value function for such a control problem. Then, extending Peng's backward semigroup method, we show the dynamic programming principle. Moreover, we prove that the value function is a viscosity solution to the following generalized Hamilton-Jacobi-Bellman equation with Dirichlet boundary condition: inf(v subset of V) {L(x,v)u(x)+f(x,u(x),del u(x)sigma(x,v),v)} = 0, x is an element of D, and u(x) = g(x), x is an element of partial derivative D, where D is a bounded set in R-d, V is a compact set in R-k, and for u is an element of C-2(D) and (x, v) is an element of D x V, L(x,v)u(x) :=1/2 Sigma(d)(i,j=1) (sigma sigma*) i,j (x, v) partial derivative(2)u/partial derivative x(i)partial derivative x(j) (x) Sigma(d)(i=1) b(i)(x, v) partial derivative u/partial derivative x(i) (x).