The recent measurement of the Higgs boson mass implies a relatively slow rise of the standard model Higgs potential at large scales, and a possible second minimum at even larger scales. Consequently, the Higgs field may develop a large vacuum expectation value during inflation. The relaxation of the Higgs field from its large postinflationary value to the minimum of the effective potential represents an important stage in the evolution of the Universe. During this epoch, the time-dependent Higgs condensate can create an effective chemical potential for the lepton number, leading to a generation of the lepton asymmetry in the presence of some large right-handed Majorana neutrino masses. The electroweak sphalerons redistribute this asymmetry between leptons and baryons. This Higgs relaxation leptogenesis can explain the observed matter-antimatter asymmetry of the Universe even if the standard model is valid up to the scale of inflation, and any new physics is suppressed by that high scale.