The main theme of the present study is to analyze numerically the effects of the magnetic field on the hybrid nanofluid flow over a flat elastic surface. The effects of the thermal and velocity slips are also analyzed in view of the hybrid nanofluid flow. It is considered a combination of titanium oxide (TiO2) and copper oxide (CuO) nanoparticles that are suspended in the incompressible and electrically conducting fluid (water). The behavior of the Brownian motion of the nanoparticles and the thermophoretic forces are contemplated in the physical and mathematical formulations. Moreover, the impact of the Joule heating and viscous dissipation are also discussed using the energy equation. The mathematical modeling is simulated with the help of similarity variables. The resulting equations are solved using the Keller–Box method with a combination of finite difference schemes (FDSs). Hybrid nanofluids provide significant advantages over the usual heat transfer fluids. Therefore, the use of nanofluids is beneficial to improve the thermophysical properties of the working fluid. All of the results are discussed for the various physical parameters involved in governing the flow. From the graphical results, it is found that the hybrid nanoparticles improve the concentration, temperature, and velocity profiles, as well as the thickness of the relevant boundary layer. The conjunction of a magnetic field and the velocity slip, strongly opposes the fluid motion. The boundary layer thickness and concentration profile are significantly reduced with the higher levels of the Schmidt number.