  Exclusive production of light vector mesons in deep inelastic scattering is
calculated at next-to-leading order in the dipole picture in the limit of high
photon virtuality. The resulting expression is free of any divergences and
suitable for numerical evaluations. The higher-order corrections are found to
be numerically important, but they can be mostly captured by the
nonperturbative fit parameters describing the initial condition for the
small-$x$ evolution of the dipole scattering amplitude. The vector meson
production cross section is shown to depend only weakly on the meson
distribution amplitude and the factorization scale. We also present
phenomenological comparisons of our result to the existing exclusive $\phi$ and
$\rho$ production data from HERA and find an excellent agreement at high
virtualities.
