The evolutionary processes that drive rapid species diversification are poorly understood. In particular, it is unclear whether Darwinian adaptation or non-adaptive processes are the primary drivers of explosive species diversifications. Here we show that repeated rapid radiations within New World lupins (Lupinus, Leguminosae) were underpinned by a major increase in the frequency of adaptation acting on coding and regulatory changes genome-wide. This contrasts with far less frequent adaptation in genomes of slowly diversifying lupins and all other plant genera analysed. Furthermore, widespread shifts in optimal gene expression coincided with shifts to high rates of diversification and evolution of perenniality, a putative key adaptation trait thought to have triggered the evolutionary radiations in New World lupins. Our results reconcile long-standing debate about the relative importance of protein-coding and regulatory evolution, and represent the first unambiguous evidence for the rapid onset of lineage- and genome-wide accelerated Darwinian evolution during rapid species diversification.