The observation of forward proton scattering in association with lepton pairs (e^{+}e^{-}+p or μ^{+}μ^{-}+p) produced via photon fusion is presented. The scattered proton is detected by the ATLAS Forward Proton spectrometer, while the leptons are reconstructed by the central ATLAS detector. Proton-proton collision data recorded in 2017 at a center-of-mass energy of sqrt[s]=13 TeV are analyzed, corresponding to an integrated luminosity of 14.6 fb^{-1}. A total of 57 (123) candidates in the ee+p (μμ+p) final state are selected, allowing the background-only hypothesis to be rejected with a significance exceeding 5 standard deviations in each channel. Proton-tagging techniques are introduced for cross-section measurements in the fiducial detector acceptance, corresponding to σ_{ee+p}=11.0±2.6(stat)±1.2(syst)±0.3(lumi) and σ_{μμ+p}=7.2±1.6(stat)±0.9(syst)±0.2(lumi) fb in the dielectron and dimuon channel, respectively.