  We report the first observation of the semi-muonic decay $D^+\to \omega
\mu^+\nu_\mu$ using an $e^+e^-$ collision data sample corresponding to an
integrated luminosity of $2.93~\mathrm{fb}^{-1}$ collected with the BESIII
detector at a center-of-mass energy of 3.773 GeV. The absolute branching
fraction of the $D^+\to \omega \mu^+\nu_\mu$ decay is measured to be ${\mathcal
B}_{D^+\to \omega\mu^+\nu_\mu}=(17.7\pm1.8_{\rm stat}\pm1.1_{\rm syst})\times
10^{-4}$. Its ratio with the world average value of the branching fraction of
the $D^+\to \omega e^+\nu_e$ decay probes lepton flavor universality and it is
determined to be ${\mathcal B}_{D^+\to \omega \mu^+\nu_\mu}/{\mathcal B}^{\rm
PDG}_{D^+\to \omega e^+\nu_e}=1.05\pm0.14$, in agreement with the standard
model expectation within one standard deviation.
