We study the Dirichlet problem for viscous Hamilton-Jacobi Equations. Despite this type of equations seems to be uniformly elliptic, loss of boundary conditions may occur because of the strong nonlinearity of the first-order part and therefore the Dirichlet boundary condition has to be understood in the sense of viscosity solutions theory. Under natural assumptions on the initial and boundary data, we prove a Strong Comparison Result which allows us to obtain the existence and the uniqueness of a continuous solution which is defined globally in time.