For multiband superconductors, the orbital multiplicity yields orbitaldifferentiation in normal-state properties and can lead to orbital-selective spin-fluctuation Cooper pairing. The orbital-selective phenomenon has becomeincreasingly pivotal in clarifying the pairing"enigma", particularly for multibandhigh-temperature superconductors. Meanwhile, in one-unit-cell (1-UC) FeSe/SrTiO3, since the standard electron-hole Fermi pocket nesting scenario isinapplicable, the actual pairing mechanism is subject to intense debate. Here, bymeasuring high-resolution Bogoliubovquasiparticle interference, we reportobservations of highly anisotropic magnetic Cooper pairing in 1-UC FeSe.Theoretically, it is important to incorporate orbitally selective effects of electroniccorrelations within a spin-fluctuation pairing calculation, where the dxyorbitalbecomes coherence-suppressed. The resulting pairing gap is compatible with theexperimentalfindings, which suggests that high-TcCooper pairing with orbitalselectivity applies to 2D-limit 1-UC FeSe. Ourfindings imply the general existence of orbital selectivity in iron-basedsuperconductors and the universal significance of electron correlations in high-Tcsuperconductors