The purpose of this study was to evaluate the potential of alginate-based packaging materials with the incorporation of protein hydrolysates to improve the safety and quality of chicken meat during storage. Physicochemical parameters, microbiological indicators, and color characteristics of chicken meat packaged in bioactive films were determined. We observed a significant increase in moisture content for samples in polyethylene films (by 10.5%) and decrease for the samples in alginate-based films by 5.3%. The highest mass losses were found for the sample without packaging material (20.4%) and for the samples wrapped in alginate films (15.9–17.9%). When packing meat samples by immersion method, a gradual decrease in weight was found (up to 9.1%). On the 7th day of storage, the pH value of the control sample reached 6.55, while for the samples in bioactive alginate-based materials pH level was 6.0–6.15. The most pronounced oxidative processes were observed in the control meat sample (5.1 mmol (12O2)/kg). The application of bioactive alginate-based films led to a significant reduction in fatty peroxide value by 56.2%. The total microbial count in the meat samples packaged in bioactive films was 3.5–5 times lower than in the control sample. Chicken meat wrapped in alginate-based films with protein hydrolysates maintains more stable color characteristics, the lightness index (L) decreased to 37.5, and the redness index (b) increased to 3.4 on the 7th day of storage.