  We perform a detailed study of the exclusive Higgs decays $h\to MZ$ and $h\to
MW$, where $M$ is a pseudoscalar or vector meson, using the QCD factorization
approach. We allow for the presence of new-physics effects in the form of
modified Higgs couplings to gauge bosons and fermions, including the
possibility of flavor-changing Higgs couplings. We show that the decays $h\to
VZ$ exhibit a strong sensitivity to the effective CP-even and CP-odd $h\gamma
Z$ couplings. When combined with a measurement of the $h\to\gamma Z$ decay
rate, this can be used to extract these couplings up to a sign ambiguity in the
CP-odd coefficient. Some of the $h\to MW$ decay modes can be used to probe for
flavor-violating Higgs couplings involving the top quark.
