The RNA world hypothesis confers a central role to RNA molecules in information encoding and catalysis. Even though evidence in support of this hypothesis has accumulated from both experiments and computational modelling, the transition from an RNA world to a world where heritable genetic information is encoded in DNA remains an open question. Recent experiments show that both RNA and DNA templates can extend complementary primers using free RNA/DNA nucleotides, either non-enzymatically or in the presence of a replicase ribozyme. Guided by these experiments, we analyse protocellular evolution with an expanded set of reaction pathways made possible through the presence of DNA nucleotides. We allow for the possibility of auto-catalytic synthesis of replicase ribozymes from DNA templates encoding the replicase, after the chance emergence of a replicase through non-enzymatic reactions. By encapsulating these reactions inside three different types of protocellular compartments, each subject to distinct modes of selection, we show how protocells containing replicase-encoded DNA in low copy numbers and replicases in high copy numbers can dominate the population. Our work suggests a plausible pathway for the transition from an RNA world to a mixed RNA-DNA world characterized by Darwinian evolution, where DNA sequences encode heritable phenotypes.