We introduce a new method for calculating the O(α3) hadronic-vacuum-polarization contribution to the muon anomalous magnetic moment from ab initio lattice QCD. We first derive expressions suitable for computing the higher-order contributions either from the renormalized vacuum polarization function Π^(q2) or directly from the lattice vector-current correlator in Euclidean space. We then demonstrate the approach using previously published results for the Taylor coefficients of Π^(q2) that were obtained on four-flavor QCD gauge-field configurations with physical light-quark masses. We obtain 1010aμHVP,HO=−9.3(1.3) in agreement with, but with a larger uncertainty than, determinations from e+e−→hadrons data plus dispersion relations.