The　classical　Melnikov　method　for　heteroclinic　orbits　is　extended　theoretically　to　a　class　of　hybrid　piecewise-smooth　systems　with　impulsive　effect　and　noise　excitation.　We　assume　that　the　unperturbed　system　is　a　piecewise　Hamiltonian　system　with　a　pair　of　heteroclinic　orbits.　The　heteroclinic　orbit　transversally　jumps　across　the　first　switching　manifold　by　an　impulsive　effect　and　crosses　the　second　switching　manifold　continuously.　In　effect,　the　trajectory　of　the　corresponding　perturbed　system　crosses　the　second　switching　manifold　by　applying　the　reset　map　describing　the　impact　rule　instantaneously.　The　random　Melnikov　process　of　such　systems　is　then　derived　by　measuring　the　distance　of　perturbed　stable　and　unstable　manifolds,　and　the　criteria　for　the　onset　of　chaos　with　or　without　noise　excitation　is　established.　In　this　derivation　process,　we　overcome　the　difficulty　that　the　derivation　method　of　the　corresponding　homoclinic　case　cannot　be　directly　used　due　to　the　difference　between　the　symmetry　of　the　homoclinic　orbit　and　the　asymmetry　of　the　heteroclinic　orbit.　Finally,　we　investigate　the　complicated　dynamics　of　a　particular　piecewise-smooth　system　with　and　without　noise　excitation　under　the　reset　maps,　impulsive　effect,　and　non-autonomous　periodic　and　damping　perturbations　by　this　new　extended　method　and　numerical　simulations.　The　numerical　results　verify　the　correctness　of　the　theoretical　results　and　demonstrate　that　this　extended　method　is　simple　and　effective　for　studying　the　dynamics　of　such　systems.