We introduce an emulator approach to predict the non-linear matter power spectrum for broad classes of beyond-$Lambda$CDM cosmologies, using only a suite of $Lambda$CDM $N$-body simulations. By including a range of suitably modified initial conditions in the simulations, and rescaling the resulting emulator predictions with analytical `halo model reactions, accurate non-linear matter power spectra for general extensions to the standard $Lambda$CDM model can be calculated. We optimise the emulator design by substituting the simulation suite with non-linear predictions from the standard {sc halofit} tool. We review the performance of the emulator for artificially generated departures from the standard cosmology as well as for theoretically motivated models, such as $f (R)$ gravity and massive neutrinos. For the majority of cosmologies we have tested, the emulator can reproduce the matter power spectrum with errors $lesssim 1%$ deep into the highly non-linear regime. This work demonstrates that with a well-designed suite of $Lambda$CDM simulations, extensions to the standard cosmological model can be tested in the non-linear regime without any reliance on expensive beyond-$Lambda$CDM simulations.
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