We present a novel approach for computing resonant inelastic X-ray scattering (RIXS) cross sections within the equation-of-motion coupled-cluster (EOM-CC) framework. The approach is based on recasting the sum-over-states expressions for RIXS moments into closed-form expressions by using damped response theory. Damped response formalism allows one to circumvent problems of divergent behavior of response equations in the resonant regime. However, the convergence of response equations in the X-ray frequency range is often erratic due to the electronically metastable (i.e., resonant) nature of the virtual core-excited states embedded in the valence ionization continuum. We circumvent this problematic behavior by extending the core-valence separation (CVS) scheme, which decouples the valence-excited and core-excited configurations of the excitation manifold, into the response domain. The accuracy of the CVS-enabled damped response theory, implemented within the EOM-EE-CCSD (EOM-CC for excitation energies with single and double excitations) framework, is assessed by comparison against standard damped EOM-EE-CCSD response calculations. The capabilities of the new approach are illustrated by calculations of RIXS cross sections for benzene and benzene radical cation.