Effects from a finite top quark mass on the H+ n-jet coss section through gluon fusion are studied for n = 0 /n ≥ 1 at NNLO/NLO QCD. For this purpose, sub-leading terms in 1 /m t are calculated. We show that the asymptotic expansion of the jet-vetoed cross section at NNLO is very well behaved and that the heavy-top approximation is valid at the five permille level up to jet-veto cuts of 300 GeV. For the inclusive Higgs+jet rate, we introduce a matching procedure that allows for a reliable prediction of the top-mass effects using the expansion in 1 /m t . The quality of the effective field theory to evaluate differential K-factors for the distribution of the hardest jet is found to be better than 1-2% as long as the transverse momentum of the jet is integrated out or remains below about 150 GeV.