We theoretically investigate a scheme to obtain sub-diffraction-limited resolution in coherent anti-Stokes Raman scattering (CARS) microscopy. We find using density matrix calculations that the rise of vibrational (Raman) coherence can be strongly suppressed, and thereby the emission of CARS signals can be significantly reduced, when pre-populating the corresponding vibrational state through an incoherent process. The effectiveness of pre-populating the vibrational state of interest is investigated by considering the excitation of a neighbouring vibrational (control) state through an intense, mid-infrared control laser. We observe that, similar to the processes employed in stimulated emission depletion microscopy, the CARS signal exhibits saturation behaviour if the transition rate between the vibrational and the control state is large. Our approach opens up the possibility of achieving chemically selectivity sub-diffraction-limited spatially resolved imaging.