Active packaging with biobased polymers aim to extend the shelf life of food and to improve the environmental sustainability of the food industry. This new concept was tested with samples of fresh poultry meat wrapped with chitosan reinforced with 2.5% of commercial nanocellulose (NC) incorporating 1% of essential oils (EO) from Aloysia citrodora (ACEO) and Cymbopogon citratus (CCEO). The performance of the bionanocomposites containing EOs was assessed and compared with unwrapped meat samples and samples wrapped with chitosan/NC, during a 13 day period of refrigerated storage for several physicochemical parameters related to food deterioration and microbial growth. Wrapping the meat with the chitosan/NC polymer helped to increase the shelf life of the meat. The incorporation of EOs added extra activity to the biocomposites, further delaying the meat deterioration process, by halting the lipid oxidation and the Enterobactereaceae growth until the 9th day. The composition of both EOs was similar, with the main components contributing to the increased activity of the biopolymers being geranial and neral. The performance of ACEO surpassed that of CCEO, namely on the Enterobactereaceae growth. This trend may be associated with ACEO’s higher phenolic content and the higher antioxidant activity of the compounds released by the ACEO biopolymers.