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We model satellite quenching at z ∼ 1 by combining 14 massive (1013.8 < Mhalo/M⊙ < 1015) clusters at 0.8 < z < 1.3 from the GOGREEN and GCLASS surveys with accretion histories of 56 redshift-matched analogues from the IllustrisTNG simulation. Our fiducial model, which is parametrized by the satellite quenching time-scale (τquench), accounts for quenching in our simulated satellite population both at the time of infall by using the observed coeval field quenched fraction and after infall by tuning τquench to reproduce the observed satellite quenched fraction versus stellar mass trend. This model successfully reproduces the observed satellite quenched fraction as a function of stellar mass (by construction), projected cluster-centric radius, and redshift and is consistent with the observed field and cluster stellar mass functions at z ∼ 1. We find that the satellite quenching time-scale is mass dependent, in conflict with some previous studies at low and intermediate redshift. Over the stellar mass range probed (M⋆ > 1010 M⊙), we find that the satellite quenching time-scale decreases with increasing satellite stellar mass from ∼1.6 Gyr at 1010 M⊙ to ∼0.6−1 Gyr at 1011 M⊙ and is roughly consistent with the total cold gas (HI + H2) depletion time-scales at intermediate z, suggesting that starvation may be the dominant driver of environmental quenching at z < 2. Finally, while environmental mechanisms are relatively efficient at quenching massive satellites, we find that the majority ($\sim 65{\!-\!}80{{\ \rm per\ cent}}$) of ultra-massive satellites (M⋆ > 1011 M⊙) are quenched prior to infall.