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 5 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.