In this study, two different cooling systems for the thermal management of a photovoltaic (PV) module were developed. A PV/thermoelectric generator (TEG) and PV/TEG-mini-channel cooling systems were considered; in the later system, water and water-based Al2O3 nanofluids were used in the cooling channel. The effective cooling of the PV module was achieved by using higher-loading nanoparticles in the base fluid, while the nanofluid exhibited a non-Newtonian behavior. The PV/TEG with a cooling channel system was numerically assessed with respect to various values of Reynolds numbers (between 5 and 250), inlet nanofluid temperatures (between 288.15 K and 303.15 K), and nanoparticle volume fractions in the base fluid (between 1% and 5%). Variations in average cell temperature, PV power, TEG power, and efficiencies were computed by varying the pertinent parameters of interest with Galerkin’s weighted residual finite element method. The most favorable case for cooling was obtained with TEG-cooling channel at φ = 5% and Re = 250. In this case, PV electrical power increased by about 8.1% and 49.2% compared to the PV/TEG and PV system without cooling, respectively. The TEG output power almost doubled when compared to the PV/TEG system for all channel models at Re = 250. The inlet temperature of the nanofluid has a profound impact on the overall efficiency and power increment of the PV module. The use of the PV/TEG-cooling channel with the lowest fluid inlet temperature (288.15 K) and nanofluid at the highest particle loading (φ = 5%) resulted in a PV efficiency increment of about 52% and 10% compared to the conventional PV system without cooling and the PV/TEG system. In this case, the TEG efficiency rises by about 51% in the PV/TEG nanofluid model compared to the PV/TEG model.