Plant bioactive compounds have antimicrobial and antioxidant activities that allow them to be used as a substitute for synthetic chemical additives in both food and food packaging. To improve its sensory and bactericidal effects, its use in the form of effective combinations has emerged as an interesting possibility in the food industry. In this study, the antimicrobial activities of essential oils (EOs) of cinnamon bark, cinnamon leaves, and clove and the pure compounds vanillin, eugenol, and cinnamaldehyde were investigated individually and in combination against Listeria monocytogenes and Escherichia coli O157:H7. The possible interactions of combinations of pure compounds and EOs were performed by the two-dimensional checkerboard assay and isobologram methods. Vanillin exhibited the lowest antimicrobial activity (MIC of 3002 ppm against L. monocytogenes and 2795 ppm against E. coli O157:H7), while clove and cinnamon bark EOs exhibited the highest antimicrobial activity (402–404 against L. monocytogenes and 778–721 against E. coli O157:H7). For L. monocytogenes, pure compound eugenol, the main component of cinnamon leaves and clove, showed lower antimicrobial activity than EOs, which was attributed to the influence of the minor components of the EOs. The same was observed with cinnamaldehyde, the main component of cinnamon bark EO. The combinations of vanillin/clove EO and vanillin/cinnamon bark EO showed the most synergistic antimicrobial effect. The combination of the EOs of cinnamon bark/clove and cinnamon bark/cinnamon leaves showed additive effect against L. monocytogenes but indifferent effect against E. coli O157:H7. For L. monocytogenes, the best inhibitory effects were achieved by cinnamon bark EO (85 ppm)/vanillin (910 ppm) and clove EO (121 ppm)/vanillin (691 ppm) combinations. For E. coli, the inhibitory effects of clove EO (104 ppm)/vanillin (1006 ppm) and cinnamon leaves EO (118 ppm)/vanillin (979 ppm) combinations were noteworthy. Some of the tested combinations increased the antimicrobial effect and would allow the effective doses to be reduced, thereby offering possible new applications for food and active food packaging.