Antimicrobial edible coatings can eliminate the risk of pathogen contamination on the surface of poultry products during storage. In this study, an edible coating (EC) based on wheat gluten, Pistacia vera L. tree resin (PVR), and the essential oil (EO) of PVR was applied on chicken breast fillets (CBF) by a dipping method to prevent the growth of Salmonella Typhimurium and Listeria monocytogenes. The samples were packed in foam trays wrapped with low-density polyethylene stretch film and stored at 8 °C for 12 days to observe the antimicrobial effects and sensory properties. The total bacteria count (TBC), L. monocytogenes, and S. Typhimurium were recorded during storage. The samples coated with EC, containing 0.5, 1, 1.5, and 2% v/v EO (ECEO), showed significant decreases in microbial growth compared to the control samples. The growth of TBC, L. monocytogenes, and S. Typhimurium was suppressed by 4.6, 3.2, and 1.6 logs, respectively, at the end of 12 days on the samples coated with ECEO (2%) compared to the uncoated controls (p < 0.05). Coating with ECEO (2%) also preserved the appearance, smell, and general acceptance parameters better than uncoated raw chicken (p < 0.05) on the fifth day of storage. In grilled chicken samples, ECEO (2%) did not significantly change the appearance, smell, and texture (p > 0.05) but increased the taste and general acceptance scores. Therefore, ECEO (2%) can be a feasible and reliable alternative to preserve CBFs without adversely affecting their sensory properties.