Non-volcanic continental passive margins have traditionally been considered to be tectonically and magmatically inactive once continental breakup has occurred and seafloor spreading has commenced. We use ambient-noise tomography to constrain Rayleigh-wave phase-velocity maps beneath the eastern Gulf of Aden (eastern Yemen and southern Oman). In the crust, we image low velocities beneath the Jiza-Qamar (Yemen) and Ashawq-Salalah (Oman) basins, likely caused by the presence of partial melt associated with magmatic plumbing systems beneath the rifted margin. Our results provide strong evidence that magma intrusion persists after breakup, modifying the composition and thermal structure of the continental margin. The coincidence between zones of crustal intrusion and steep gradients in lithospheric thinning, as well as with transform faults, suggests that magmatism post-breakup may be driven by small-scale convection and enhanced by edge-driven flow at the juxtaposition of lithosphere of varying thickness and thermal age.