The magnetocaloric effect in exchange-coupled synthetic-antiferromagnet multilayers is investigated experimentally and theoretically. We observe a temperature-controlled inversion of the effect, where the entropy increases on switching the individual ferromagnetic layers from anti-parallel to parallel alignment near their Curie point. Using a microscopic analytical model as well as numerical atomistic-spin simulations of the system, we explain the observed effect as due to the interplay between the intra- and inter-layer exchange interactions, which either add up or counteract to effectively modulate the Curie temperature of the dilute ferromagnetic layers. The proposed method of designing tunable, strongly magneto-caloric materials should be of interest for such applications as heat-assisted spintronics and magnetic refrigeration.