Today, mango by-products are produced in significant amounts, posing an environmental problem. Their incorporation into food products offers a solution to this problem. An extruded food product formulated with the flours of white corn, mango peel, and mango kernel was developed. To obtain the desired quality regarding the physical and chemical characteristics of the extruded food product, an optimization study was carried out. Response surface methodology was used to evaluate the effect of the following extrusion process variables on the physical (expansion index (EI), hardness, water absorption index (WAI), water solubility index (WSI)) and chemical properties (total polyphenol content and antioxidant capacity): the die temperature (DT, 100–130 °C), the feed moisture content (FMC, 17–21%), and the screw speed (SS, 80–120 rpm). Response surface and regression models were performed to determine the responses as a function of the process variables. Model optimization was carried out with an R2 of >0.60, maximizing the WAI and minimizing the hardness and the WSI. The optimal conditions were a DT of 120.66 °C, an FMC of 21.88%, and an SS of 66.36 rpm. The extruded product’s characteristics were an EI of 1.10, a hardness of 63.66 N, a WAI of 5.41 g/g, a WSI of 16.20%, a TPC of 3402 mg GAE/100 g sample, and an antioxidant capacity of 90.09 mg Etrolox/g (measured by DPPH) and 79.38 mg Etrolox/g (measured by ABTS); the overall desirability value was 0.870.