Essential oils (EOs) are plant mixtures that are known to present strong bioactivities, including a wide antimicrobial action. Biofilms are microbial sessile structures that represent the default mode of growth of microorganisms in most environments. This study focused on the antimicrobial action of the EO extracted from one of the most representative oregano species, that is, Origanum vulgare (subsp. hirtum), against two important foodborne pathogens, Salmonella enterica (serovar Typhimurium) and Listeria monocytogenes. For this, the minimum inhibitory concentrations of the EO against the planktonic and biofilm growth of each bacterium were determined (MICs, MBICs), together with the minimum bactericidal and biofilm eradication concentrations (MBCs, MBECs). The EO was also analyzed for its chemical composition by gas chromatography—mass spectrometry analysis (GC-MS). The influence of EO exposure on the expression of some important virulence genes (hly, inlA, inlB and prfA) was also studied in L. monocytogenes. Results revealed a strong antibacterial and antibiofilm action with MICs and MBICs ranging from 0.03% to 0.06% (v/v) and from 0.06% to 0.13% (v/v), respectively. The application of the EO at 6.25% (v/v) for 15 min resulted in the total eradication of the biofilm cells of both pathogens. The EO was mainly composed of thymol, p-cymene, γ-terpinene and carvacrol. The 3 h exposure of L. monocytogenes planktonic cells to the EO at its MBIC (0.06% v/v) resulted in the significant downregulation of all the studied genes (p < 0.05). To sum, the results obtained advocate for the further exploitation of the antimicrobial action of oregano EO in food and health applications.