We consider the transverse-momentum distribution of a Higgs boson produced through gluon fusion in hadron collisions. At small transverse momenta, the large logarithmic terms are resummed up to next-to-leading-logarithmic (NLL) accuracy. The resummed computation is consistently matched to the next-to-leading-order (NLO) result valid at large transverse momenta. The ensuing Standard Model prediction is supplemented by possible new-physics effects parametrised through three dimension-six operators related to the modification of the top and bottom Yukawa couplings, and to the inclusion of a point-like Higgs-gluon coupling, respectively. We present resummed transverse-momentum spectra including the effect of these operators at NLL+NLO accuracy and study their impact on the shape of the distribution. We find that such modifications, while affecting the total rate within the current uncertainties, can lead to significant distortions of the spectrum. The proper parametrization of such effects becomes increasingly important for experimental analyses in Run II of the LHC.