The observation of forward proton scattering in association with lepton pairs (e(+)e(-) + p or mu(+)mu(-) + 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 root 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 (mu mu + p) final state arc 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 sigma(ee)(+p) = 11.0 +/- 2.6(stat) 1.2(syst) +/- 0.3(lumi) and sigma(mu)(mu+)(p) = 7.2 +/- 1.6(stat) +/- 0.9(syst) 0.2(lumi) fb in the dielectron and dimuon channel, respectively.