The interferon response is a potent antiviral defense mechanism, but its effectiveness depends on its timing relative to viral replication. Here, we report viral replication and host response kinetics in patients at the start of SARS-CoV-2 infection and explore the impact of these kinetics experimentally. In both longitudinal patient nasopharyngeal samples and airway epithelial organoids, we found that SARS-CoV-2 initially replicated exponentially with a doubling time of ~6hr, and induced interferon stimulated genes (ISGs) with delayed timing relative to viral replication. Prior exposure to rhinovirus increased ISG levels and blocked SARS-CoV-2 replication. Conversely, inhibiting ISG induction abrogated interference by rhinovirus and enhanced SARS-CoV-2 replication rate. These results demonstrate the importance of initial interferon-mediated defenses in determining the extent to which SARS-CoV-2 can replicate at the start of infection and indicate that biological variables that alter the airway interferon response, including heterologous induction of innate immunity by other viruses, could profoundly impact SARS-CoV-2 susceptibility and transmission.