Background: SARS-CoV-2 infection in pregnant women may induce inflammation within the amniotic cavity and/or an increase in proinflammatory cytokines in fetal circulation. The aim was to investigate levels of IL-6 in maternal blood, umbilical cord blood, and amniotic fluid in pregnant women infected with SARS-CoV-2 at delivery. Methods: A single-center prospective observational case–control study of pregnant women diagnosed with SARS-CoV-2 infection at delivery was conducted. A total of 48 infected and 42 healthy women had IL-6 concentrations measured in their blood, amniotic fluid, and umbilical cord blood. Results: The concentrations of IL-6 in maternal blood and amniotic fluid were similar in the study and control groups, while umbilical cord blood concentrations were significantly higher in SARS-CoV-2-positive women. The umbilical cord blood IL-6 concentration was related to composite neonatal morbidity. Conclusions: Maternal SARS-CoV-2 infection in pregnant women at delivery increases umbilical cord blood IL-6 concentration. The correlation between maternal and umbilical blood concentrations indicates a possibility of passage of IL-6 through the placenta. Perinatal alterations resulting from maternal SARS-CoV-2 infection at delivery carry a risk of impacting the health of infants even in asymptomatic course of infection.