A search for dark matter is performed using events with large missing transverse momentum and a Higgs boson decaying either to a pair of bottom quarks or to a pair of photons. The data from proton-proton collisions at a center-of-mass energy of 13 TeV, collected with the CMS detector at the LHC, correspond to an integrated luminosity of 2.3 fb$^{-1}$. Results are interpreted in the context of a Z'-two-Higgs-doublet model, where a high-mass resonance Z' decays into a pseudoscalar boson A and a CP-even scalar Higgs boson, and the A decays to a pair of dark matter particles. No significant excesses are observed over the background prediction. Combining results from the two decay channels yields exclusion limits in the signal cross section in the $m_{\mathrm{Z'}} $-$ m_{\textrm{A}}$ phase space. The observed data exclude, for Z' coupling strength $g_{\mathrm{Z'}} = $ 0.8 and $m_{\mathrm{A}} = $ 300 GeV for example, the Z' mass range of 600 to 1860 GeV. This is the first result on a search for dark matter produced in association with a Higgs boson that includes constraints on $\mathrm{h} \to γγ $ obtained at $ \sqrt{s} = $ 13 TeV