Many extensions of the Standard Model predict the production of Dark Matter particles in association with Higgs bosons. This search examines the final state of missing transverse momentum accompanied by a $b\bar{b}$ pair originating from a Higgs boson decay. For this purpose, proton-proton collision data are used, which is produced at 13 TeV center-of-mass energy and recorded by the ATLAS experiment at the LHC, amounting to an integrated luminosity of $139\, \mathrm{fb}^{-1}$. The increase in integrated luminosity in conjunction with several analysis optimizations result in a better sensitivity in comparison to previous iterations of the analysis. No significant deviation from the Standard Model is observed and the results are interpreted in the context of the Two-Higgs-Doublet models extended with an additional vector or pseudoscalar mediator.