OK, I don't know how I did it but I misread your question as being why all the non-mammalian Synapsids went extinct. I answered your question at the end, but I'm leaving the rest as-is until I figure out how to rewrite it (it was rather too much work to delete).
According to The Wikipedia page for Synapsids, the answer really depends on the Synapsis since there are quite a gap between the start of the Synapsid clade and the appearance of Cynodonts.
The first big "culling" of Synapsids (i.e. the extinction of all groups except one clade that led to mammals - Therapsids in this case) seems to occurred over the Permian era. Some groups went extinct in the early Permian but most groups went extinct at various points during the Middle Permian. No clear reason is given for these various extinctions (though the page for Dinocephalia suggests "disease, sudden climatic change, or other factors of environmental stress may have brought about their end"... which basically covers all possible extinction reasons so that's helpful); it's probably various reasons specific to each. Most species do go extinct after all.
Among Therapsids, the only clade that survived the Middle Permian, most seem to have been wiped out in the Permian-Triassic extinction, the largest extinction event in Earth's history, of which only three Therapsid clades survived: Therocephalians, which went extinct soon after, Dicynodonts, which were extinct by the Early Cretaceous, and Cynodonts. Aside from the usual environmental factors, a reason for the extinction of those two clades could have been competition from the then-ascendant Dinosaurs, and Cynodonts themselves.
As for when the last non-mammal Synapsid when extinct, it appears to have been a member of the Tritylodonts, which survived into the Cretaceous (and arguably having a representative after the K/Pg extinction, emphasis on "arguably"). Mind you these are very close cousins to Mammals indeed:
Tritylodontids ("three knob teeth", named after the shape of animal's teeth) were small to medium-sized, highly specialized and extremely mammal-like cynodonts, bearing several mammalian hallmarks like erect limbs, endothermy and lactation. They were the last known family of the non-mammalian synapsids.