  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.
