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 √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 five standard deviations in each channel. Proton-tagging techniques are introduced for cross-section measurements in the fiducial detector acceptance, corresponding to σee+p = 11.0 ± 2.6(stat) ± 1.2(syst) ± 0.3 (lumi) and σμμ+p = 7.2 ± 1.6(stat) ± 0.9(syst) ± 0.2(lumi) fb in the dielectron and dimuon channel, respectively.