In addition to explaining the masses of elementary particles, the Higgs boson may have far-reaching implications for the generation of the matter content in the Universe. For instance, the Higgs boson plays a key role in two main theories of baryogenesis, namely, electroweak baryogenesis and leptogenesis. In this Letter, we propose a new cosmological scenario where the Higgs chemical potential mediates asymmetries between visible and dark matter sectors, either generating a baryon asymmetry from a dark matter asymmetry or vice versa. We illustrate this mechanism with a simple model with two new fermions coupled to the Higgs boson and discuss the associated signatures.