The study aimed to determine the effect of dietary inclusion of the recommended (6.5 mg Cu/kg diet) or double the recommended (13.0 mg Cu/kg diet) levels of copper nanoparticles (CuNPs) in combination with different types of dietary fibre on selected redox status indicators in the blood and tissues of male Wistar rats. Control groups were fed diets containing cellulose and a mineral mixture with standard or enhanced content of CuCO3. The experimental groups were fed a diet supplemented with CuNPs (6.5 or 13 mg/kg) and combined with various fibre types—cellulose, pectin, inulin, or psyllium. After the feeding period, rats’ organs were collected to assess selected indicators of redox status. The obtained results suggest that the addition of dietary fibre in the form of inulin may beneficially stimulate the response of the redox system in the conditions of CuNPs nutrition at the recommended dose, pectin, or psyllium in the case of an excessive supply of CuNPs in the diet. Thus, selecting the appropriate type of dietary fibre based on the CuNPs’ level in the diet may effectively protect the organism from the potentially harmful prooxidative effect of CuNPs, ultimately contributing to a favourable regulation of their metabolic impact in the body.