We study the transverse momentum (qT ) spectrum of the Higgs boson produced via the annihilation of heavy quarks (s,c,b ) in proton-proton collisions. Using soft-collinear effective theory (SCET) and working in the five-flavour scheme, we provide predictions at three-loop order in resummed perturbation theory (N3LL′). We match the resummed calculation to full fixed-order results at next-to-next-to-leading order (NNLO), and introduce a decorrelation method to enable a consistent matching to an approximate N3LO (aN3LO) result. Since the b-quark initiated process exhibits large nonsingular corrections, it requires special care in the matching procedure and estimation of associated theoretical uncertainties, which we discuss in detail. Our results constitute the most accurate predictions to date for these processes in the small qT region and could be used to improve the determination of Higgs Yukawa couplings from the shape of the measured Higgs qT spectrum.