The ability to evolve is a key characteristic that distinguishes living things from non-living chemical compounds. The construction of an evolvable cell-like system entirely from non-living molecules has been a major challenge. Here we construct an evolvable artificial cell model from an assembly of biochemical molecules. The artificial cell model contains artificial genomic RNA that replicates through the translation of its encoded RNA replicase. We perform a long-term (600-generation) replication experiment using this system, in which mutations are spontaneously introduced into the RNA by replication error, and highly replicable mutants dominate the population according to Darwinian principles. During evolution, the genomic RNA gradually reinforces its interaction with the translated replicase, thereby acquiring competitiveness against selfish (parasitic) RNAs. This study provides the first experimental evidence that replicating systems can be developed through Darwinian evolution in a cell-like compartment, even in the presence of parasitic replicators.