Rifting and magmatism are fundamental geological processes that shape the surface of our planet. A relationship between the two is widely acknowledged but its precise nature has eluded geoscientists and remained controversial. Largely on the basis of detailed observations from the North Atlantic Ocean, mantle temperature was identified as the primary factor controlling magmatic production, with most authors seeking to explain observed variations in volcanic activity at rifted margins in terms of the mantle temperature at the time of break-up. However, as more detailed observations have been made at other rifted margins worldwide, the validity of this interpretation and the importance of other factors in controlling break-up style have been much debated. One such observation is from the northwest Indian Ocean, where, despite an unequivocal link between an onshore flood basalt province, continental break-up and a hot-spot track leading to an active ocean island volcano, the associated continental margins show little magmatism. Here we reconcile these observations by applying a numerical model that accounts explicitly for the effects of earlier episodes of extension. Our approach allows us to directly compare break-up magmatism generated at different locations and so isolate the key controlling factors. We show that the volume of rift-related magmatism generated, both in the northwest Indian Ocean and at the better-known North Atlantic margins, depends not only on the mantle temperature but, to a similar degree, on the rift history. The inherited extensional history can either suppress or enhance melt generation, which can explain previously enigmatic observations.