  We calculate the complete one-loop electroweak corrections to the inclusive
$H\rightarrow ZZ$ and $H\rightarrow Z\gamma$ decays in the dimension-$6$
extension of the Standard Model Effective Field Theory (SMEFT). The corrections
to $H\rightarrow ZZ$ are computed for on-shell $Z$ bosons and are a precursor
to the physical $H\rightarrow Z f {\overline{f}}$ calculation. We present
compact numerical formulas for our results and demonstrate that the logarithmic
contributions that result from the renormalization group evolution of the SMEFT
coefficients are larger than the finite NLO contributions to the decay widths.
As a by-product of our calculation, we obtain the first complete result for the
finite corrections to $G_\mu$ in the SMEFT.
