The paper deals with the steady-state stationary response of single-degree-of-freedom nonlinear secondary oscillators in stochastically driven linear primary structures. Equations governing the primary-secondary dynamic interaction are presented, followed by a concise exposition of a two-degree-of-freedom system assembly with dimensionless coefficients. Stochastic forcing is successively modelled as monochromatic, white noise, and filtered white noise process, representing extreme forms of narrow-band and wide-band excitations, as well as an intermediate case, characterised by the Clough-Penzien stationary power spectrum, commonly adopted in earthquake engineering applications. A decomposition-synthesis approach is presented in order to analytically approximate the response of the secondary system in cascade, by quantifying the statistics to individual forcing components, and synthesising the results to obtain the response of an equivalent linear secondary system. Demonstrated to the case of a Duffing secondary oscillator, and compared with pertinent Monte-Carlo simulations, the derived formulas permit accurate and efficient quantification of the secondary system's response as a function of the design input parameters of the system assembly and the excitation.