Nonlinear optical processes associated with even-order nonlinear susceptibilities are critical for both classical and quantum technologies. Inversion symmetry, however, prevents nonlinear optical responses mediated by even-order susceptibilities in several material systems that are pertinent for applications in nanophotonics. Here, we demonstrate induced nonlinear optical processes, namely, second- and fourth-harmonic generation that are naturally forbidden in an inversion symmetric system, by coupling to a photon mode of an optical cavity. For the coupled system, we control the inversion symmetry breaking by changing the light–matter coupling strength, which at the same time allows tuning of the nonlinear conversion efficiency. We find that the harmonic generation yield can be significantly increased by increasing the light–matter coupling strength in an experimentally feasible way. In addition, we find that the harmonic conversion efficiency is increased in the cavity-induced setting as opposed to using intense pump fields. Our work constitutes a step forward in the direction of realizing physically forbidden nonlinear optical processes in centrosymmetric materials widely adopted for applications in integrated photonics.