This study presents the very first report on the in vitro antiviral activity of selected essential oils of Lamiaceae plant species and their monoterpenes against severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Nineteen essential oils were obtained by hydrodistillation of dried plant material, and their monoterpene profiles were determined. In addition, the exact concentrations of each monoterpene that were found at a significant level were defined. Both essential oils and their monoterpene components were tested for cytotoxic and antiviral activity against SARS-CoV-2 in infected Vero 76 cells. The results showed that the essential oils of four Mentha species, i.e., M. aquatica L. cv. Veronica, M. pulegium L., M. microphylla K.Koch, and M. x villosa Huds., but also Micromeria thymifolia (Scop.) Fritsch and Ziziphora clinopodioides Lam., and five different monoterpenes, i.e., carvacrol, carvone, 1,8-cineol, menthofuran, and pulegone, inhibited the SARS-CoV-2 replication in the infected cells. However, the antiviral activity varied both among essential oils and monoterpenes. Carvone and carvacrol exhibited moderate antiviral activity with IC50 concentrations of 80.23 ± 6.07 μM and 86.55 ± 12.73 μM, respectively, while the other monoterpenes were less active (IC50 > 100.00 μM). Structure-activity relations of related monoterpenes showed that the presence of keto and hydroxyl groups is associated with the activity of carvone and carvacrol, respectively. Furthermore, the carvone-rich essential oil of M. x villosa had the greatest activity among all active essential oils (IC50 127.00 ± 4.63 ppm) while the other active oils exhibited mild (140 ppm < IC50 < 200 ppm) to weak antiviral activity (IC50 > 200 ppm). Both essential oils and monoterpenes showed limited or no cytotoxicity against Vero 76 cells. Hierarchical cluster analysis showed that the differences in the antiviral activity of essential oils were directly attributed to the antiviral efficacies of their particular single monoterpenes. The findings presented here on the novel antiviral property of plant essential oils and monoterpenes might be used in the development of different measures against SARS-CoV-2.