This analysis presents an observation of the photon-fusion production of $W$ boson pairs using the CMS detector at the LHC. The total cross section of the $W^+W^-$ production in photon fusion is measured using proton-proton collision data with an integrated luminosity of $138\ \mathrm{fb^{-1}}$ collected with the CMS detector in 2016--2018 at a center-of-mass energy of $\sqrt{s}=13\ \mathrm{TeV}$. Events are selected in the final state with one isolated electron and one isolated muon, and no additional tracks associated with the electron-muon production vertex. The total and fiducial production cross sections are $643^{+82}_{-78}\ \mathrm{fb}$ and $3.96^{+0.53}_{-0.51}\ \mathrm{fb}$, respectively, in agreement with the standard model predictions of $631\pm 126\ \mathrm{fb}$ and $3.87\pm 0.77\ \mathrm{fb}$. This agreement enables stringent constraints to be imposed on anomalous quartic gauge couplings within a dimension-8 effective field theory framework.