By employing a nonlocal perturbation approach and the method of invariant sets of descending flow, this manuscript investigates the existence and multiplicity of sign-changing solutions to a class of semilinear Kirchhoff equations in the following form - a + b ∫ R 3 | ∇ u | 2 Δ u + V ( x ) u = f ( u ) , x ∈ R 3 , where a , b > 0 are constants, V ∈ C ( R 3 , R ) , f ∈ C ( R , R ) . The methodology proposed in the current paper is robust, in the sense that, neither the monotonicity condition on f nor the coercivity condition on V is required. Our result improves the study made by Deng et al. (J Funct Anal 269:3500–3527, 2015), in the sense that, in the present paper, the nonlinearities include the power-type case f ( u ) = | u | p - 2 u for p ∈ ( 2 , 4 ) , in which case, it remains open in the existing literature whether there exist infinitely many sign-changing solutions to the problem above. Moreover, energy doubling is established, namely, the energy of sign-changing solutions is strictly larger than two times that of the ground state solutions for small b > 0 .