A large ensemble of experiments with an efficient climate model is carried out to examine stability of the oceanic thermohaline circulation (THC) as a function of two key processes that maintain high Atlantic salinities: the “Atmospheric Bridge” by which moisture is exported from the Atlantic to the Pacific; and “Agulhas Leakage” of salty Indian Ocean waters into the Atlantic. We find that irreversible THC collapse during the next millennium is five times more likely if Agulhas Leakage dominates over the Atmospheric Bridge. This finding is consistent with freshwater import to the Atlantic sector associated with the overturning circulation, when the Atmospheric Bridge dominates. In contrast, slight freshwater export is associated with the overturning circulation under strong Agulhas Leakage, helping to maintain higher Atlantic salinity. Predictions of future climate change therefore critically depend on better understanding of the relative importance of the Atmospheric Bridge and Agulhas Leakage.