The purpose of this study was to investigate the effect of the addition of graphene (G) and graphene oxide (GO) to polyethylene film on the lipid oxidation level and sensory quality of African catfish fillets during refrigerated storage. The study was conducted on de-skinned fillets from a local farm. A composite film with 0.1%, 0.6%, and 1% G nanoplatelets and 0.1% GO nanoplatelets was used to package the fillets. The film was obtained using the free-blowing vertical upward method. The fillets were stored in polyethylene bags at 4 °C, and tests were conducted on the day of packaging and after 3, 7, 10, and 14 days of refrigerated storage. The peroxide number, anisidine number, and Totox index were determined in the extracted lipids using the Bligh and Dyer method. Sensory analysis of the fillets was carried out using the spot method and sensory profiling. Global migration for the film was also investigated. After 14 days, fillets packaged in the plain film exhibited, on average, 50%–100% higher lipid oxidation indices than those packaged in the film with G and GO. The film with 1% G and 0.1% GO additive showed the best properties for retarding the formation of aldehydes and ketones in lipids during fillet storage. There was no significant effect of the addition of nanomaterials to the film on the sensory quality of the fillets. The global migration did not exceed the allowed limit of 10 mg/dm2 for any of the films. Composite films with nanomaterials have great potential as packaging in slowing down oxidation processes in foods.