The existence of Dark Matter is inferred from several astrophysical and cosmological observations. Several extensions of the Standard Model accommodating the present theoretical and experimental Dark Matter constraints predict associated production of Dark Matter particles with the Standard Model Higgs boson. A search for such models is presented for the final state with large missing momentum and $b$-jets from the Higgs boson decay. The results are based on 36.1$~\mathrm{fb}^{-1}$ of $pp$ collision data at a center-of-mass energy of 13 TeV recorded by the ATLAS detector at the Large Hadron Collider. An optimized event selection together with an improved Higgs boson identification in boosted topologies leads to a significant improvement in sensitivity compared to previous searches. The observed data are in good agreement with the Standard Model predictions. Thus, limits are placed on the cross-section for Dark Matter particle production in association with a Higgs boson, assuming a simplified theory model with two Higgs doublets and a heavy vector mediator. In addition, stringent model-independent limits are placed at the detector level on the production cross-section of processes beyond the Standard Model. These can be reinterpreted in the context of a wider range of theoretical models with the same event signature.