ABSTRACT We present a general method to compute the non-linear matter power spectrum for dark energy (DE) and modified gravity scenarios with per cent-level accuracy. By adopting the halo model and non-linear perturbation theory, we predict the reaction of a lambda cold dark matter (ΛCDM) matter power spectrum to the physics of an extended cosmological parameter space. By comparing our predictions to N-body simulations we demonstrate that with no-free parameters we can recover the non-linear matter power spectrum for a wide range of different w0–wa DE models to better than 1 per cent accuracy out to k ≈ 1 $h \,{\rm Mpc}^{-1}$. We obtain a similar performance for both DGP and f(R) gravity, with the non-linear matter power spectrum predicted to better than 3 per cent accuracy over the same range of scales. When including direct measurements of the halo mass function from the simulations, this accuracy improves to 1 per cent. With a single suite of standard ΛCDM N-body simulations, our methodology provides a direct route to constrain a wide range of non-standard extensions to the concordance cosmology in the high signal-to-noise non-linear regime.