Controlling the infectivity of respiratory RNA viruses is critical, especially during the current SARS-CoV-2 pandemic. There is an unmet need for therapeutic agents that can reduce viral replication, preferably independent of the accumulation of viral mutations. Zinc ions have an apparent activity as modulators of intracellular viral RNA replication and thus, appear attractive in reducing viral RNA load and infectivity. However, the intracellular concentration of zinc is usually too low for achieving an optimal inhibitory effect. Various herbal polyphenols serve as excellent zinc ionophores with known antiviral properties. Here, we combined zinc picolinate with a collection of flavonoids, representing commonly used polyphenols. Copper was added to avoid ionic imbalance during treatment and to improve efficacy. Each component separately, as well as their combinations, did not interfere with the viability of cultured A549, H1299, or Vero cells in vitro as determined by MTT assay. The safe combinations were further evaluated to determine antiviral activity. Fluorescence-activated cell sorting and quantitative polymerase chain reaction were used to evaluate antiviral activity of the combinations. They revealed a remarkable (50–95%) decrease, in genome replication levels of a diverse group of respiratory RNA viruses, including the human coronavirus OC43 (HCoV-OC43; a betacoronavirus that causes the common cold), influenza A virus (IAV, strain A/Puerto Rico/8/34 H1N1), and human metapneumovirus (hMPV). Collectively, our results offer an orally bioavailable therapeutic approach that is non-toxic, naturally sourced, applicable to numerous RNA viruses, and potentially insensitive to new mutations and variants.