Among the key characteristics of living systems are their ability to self‐replicate and the fact that they exist in an open system away from equilibrium. Herein, we show how the outcome of the competition between two self‐replicators, differing in size and building block composition, is different depending on whether the experiments are conducted in a closed vial or in an open and out‐of‐equilibrium replication–destruction regime. In the closed system, the slower replicator eventually prevails over the faster competitor. In a replication‐destruction regime, implemented through a flow system, the outcome of the competition is reversed and the faster replicator dominates. The interpretation of the experimental observations is supported by a mass‐action‐kinetics model. These results represent one of the few experimental manifestations of selection among competing self‐replicators based on dynamic kinetic stability and pave the way towards Darwinian evolution of abiotic systems.