A novel search is reported for high-mass resonances decaying to a jet and a Lorentz-boosted dijet resonance, thus denoted as trijet resonances. The search is based on data collected with the CMS detector at the LHC in proton-proton collisions at $\sqrt{s}=13~\mathrm{TeV}$, corresponding to an integrated luminosity of $138~fb^{-1}$. The boosted dijet resonance is reconstructed as a single wide jet with substructure consistent with a two-body decay. The high-mass resonance is thus considered as a dijet system. The jet substructure information and the kinematics of cascade resonance decays are exploited to disentangle the signal from the large quantum chromodynamics multijet background. The dijet mass spectrum is analyzed for the presence of new high-mass resonances, and is found to be consistent with the standard model background predictions. Results are interpreted in a warped extra dimension model where the high-mass resonance is a Kaluza-Klein gluon, the boosted dijet resonance is a radion, and final state partons are all gluons. Limits on the production cross section are set as a function of the Kaluza-Klein gluon and radion masses. These limits exclude at $95\%$ confidence level models with Kaluza-Klein gluon masses in the between 2.0 and $4.3~\mathrm{TeV}$ and radion masses in the range 0.20 and $0.74~\mathrm{TeV}$. By exploring a novel experimental signature, the discovery potential of this new physics benchmark model is significantly extended compared to previous searches at the LHC.