In agriculture, there are various types of progress, and trends in agricultural production include the minimized use of chemicals and the production of food with favorable taste and increased nutritional value. In this study, during a three-year field experiment on potato cultivar Agria, ozone was applied in a fumigation process with an invented device, and hydrogen peroxide was applied as a foliar spray. The yield of ozone-treated plants was lower than of the control plants. Hydrogen peroxide treatment did not affect the yield of potatoes. Ozone fumigation had a positive effect on the chemical composition of tubers, while in the case of hydrogen peroxide, this effect occurred only in one year of research and only for selected parameters. The antioxidant response of plants, which is a defense response to artificially induced oxidative stress, contributes to the production of antioxidant-like metabolites by increasing the antiradical activity of crops by 3–23%, increasing the total content of polyphenolic compounds by 12–108%, and increasing the total antioxidant capacity by 23–42%. The obtained results confirmed that appropriate doses of ROS can be used in the field cultivation of potatoes to increase the nutritional value of tubers.