A sanitary cover provides an improved way to prevent or lessen the spread of infectious diseases, such as COVID-19, the flu, and/or the common cold, through indirect contact with beverage containers. The sanitary cover includes a sleeve of flexible absorptive fabric that is impregnated with a liquid germicidal disinfectant. The sleeve is sized and shaped to fit onto and over the pour-through opening of a beverage container. A set of connected sanitary covers is formed by an elongate tubular sleeve of the flexible absorptive fabric impregnated with a liquid germicidal disinfectant and divided into individual ones of the sanitary covers by frangible line and a seam formed in the elongate tubular sleeve between each adjacent pair of the sanitary covers.