The first observation of electroweak production of a photon in association with two forward jets in proton-proton collisions is presented. The measurement uses data recorded by the CMS experiment at the LHC during 2016$-$2018 at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 138 fb$^{-1}$. The analysis is performed in a region enriched in photon production via vector boson fusion, with a requirement on the transverse momentum of the photon to exceed 200 GeV. The cross section is measured to be 202$^{+36}_{-32}$ fb, at a significance with respect to the null hypothesis that exceeds five standard deviations. This is in agreement with the standard model prediction of 177$^{+13}_{-12}$ fb. Differential cross sections are measured as a function of various observables. Limits are set on dimension-6 effective field theory operators that contribute to the WW$γ$ interaction. The observed 95% confidence intervals for the corresponding Warsaw basis Wilson coefficients $c_\mathrm{W}$ and $c_\mathrm{HWB}$ are [$-$0.11, 0.16] and [$-$1.6, 1.5], respectively.
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