Enhancers are fast-evolving genomic sequences that control spatiotemporal gene expression patterns. By examining enhancer turnover across mammalian species and in multiple tissue types, we uncover a relationship between the emergence of enhancers and genome organization as a function of germline DNA replication time. While enhancers are most abundant in euchromatic regions, enhancers emerge almost twice as often in late compared to early germline replicating regions, independent of transposable elements. Using a deep learning sequence model, we demonstrate that new enhancers are enriched for mutations that alter transcription factor (TF) binding. Recently evolved enhancers appear to be mostly neutrally evolving and enriched in eQTLs. They also show more tissue specificity than conserved enhancers, and the TFs that bind to these elements, as inferred by binding sequences, also show increased tissue-specific gene expression. We find a similar relationship with DNA replication time in cancer, suggesting that these observations may be time-invariant principles of genome evolution. Our work underscores that genome organization has a profound impact in shaping mammalian gene regulation.