This paper reports the observation of top-quark pair production in proton-lead collisions in the ATLAS experiment at the Large Hadron Collider. The measurement is performed using 165 nb$^{−1}$ of p+Pb data collected at $ \sqrt{s_{\textrm{NN}}} $ = 8.16 TeV in 2016. Events are categorised in two analysis channels, consisting of either events with exactly one lepton (electron or muon) and at least four jets, or events with two opposite-charge leptons and at least two jets. In both channels at least one b-tagged jet is also required. Top-quark pair production is observed with a significance over five standard deviations in each channel. The top-quark pair production cross-section is measured to be $ {\sigma}_{t\overline{t}}=58.1\pm 2.0{\left(\textrm{stat}.\right)}_{-4.4}^{+4.8}\left(\textrm{syst}.\right) $ nb, with a total uncertainty of 9%. In addition, the nuclear modification factor is measured to be $ {R}_{p\textrm{A}}=1.090\pm 0.039{\left(\textrm{stat}.\right)}_{-0.087}^{+0.094}\left(\textrm{syst}.\right) $. The measurements are found to be in good agreement with theory predictions involving nuclear parton distribution functions.[graphic not available: see fulltext]