The observation of forward proton scattering in association with lepton pairs (${e}^{+}{e}^{\ensuremath{-}}+p$ or ${\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}+p$) 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 $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ are analyzed, corresponding to an integrated luminosity of $14.6\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. A total of 57 (123) candidates in the $ee+p$ ($\ensuremath{\mu}\ensuremath{\mu}+p$) 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 ${\ensuremath{\sigma}}_{ee+p}=11.0\ifmmode\pm\else\textpm\fi{}2.6(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}1.2(\mathrm{syst})\ifmmode\pm\else\textpm\fi{}0.3(\mathrm{lumi})$ and ${\ensuremath{\sigma}}_{\ensuremath{\mu}\ensuremath{\mu}+p}=7.2\ifmmode\pm\else\textpm\fi{}1.6(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.9(\mathrm{syst})\ifmmode\pm\else\textpm\fi{}0.2(\mathrm{lumi})\text{ }\text{ }\mathrm{fb}$ in the dielectron and dimuon channel, respectively.