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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 anintegrated luminosity of $2.93~\mathrm{fb}^{-1}$ collected with the BESIIIdetector at a center-of-mass energy of 3.773 GeV. The absolute branchingfraction of the $D^+\to \omega \mu^+\nu_\mu$ decay is measured to be ${\mathcalB}_{D^+\to \omega\mu^+\nu_\mu}=(17.7\pm1.8_{\rm stat}\pm1.1_{\rm syst})\times10^{-4}$. Its ratio with the world average value of the branching fraction ofthe $D^+\to \omega e^+\nu_e$ decay probes lepton flavor universality and it isdetermined to be ${\mathcal B}_{D^+\to \omega \mu^+\nu_\mu}/{\mathcal B}^{\rmPDG}_{D^+\to \omega e^+\nu_e}=1.05\pm0.14$, in agreement with the standardmodel expectation within one standard deviation.

Observation of the semi-muonic decay $D^+\to \omega\mu^+\nu_\mu$