<p>Hybrid nanofluids are studied extensively for their better heat transfer and are used in controlled cooling systems. In real systems, the combined effects of non-Newtonian behavior, magnetic forces, porous resistance, and finite heat relaxation are important in systems where classical Fourier heat conduction and Newtonian fluid assumptions are not sufficient. This study examines the unsteady magnetohydrodynamic flow of an Eyring–Powell hybrid nanofluid (Al₂O₃–Au/water) over a stretching surface. The model includes buoyancy force, an inclined magnetic field, thermal radiation, porosity, viscous dissipation, Stefan blowing/suction, Joule heating, velocity slip, and Cattaneo–Christov heat flux. The governing PDEs are transformed into a coupled nonlinear ODE system by similarity transformations, then solved numerically using the MATLAB bvp4c solver. The numerical method is validated by comparison with published results, and the results match well. The results show that increasing the Eyring–Powell parameter from 0.5 to 1.5 reduces both the velocity and temperature profiles. A stronger magnetic field and higher porosity reduce velocity due to Lorentz and Darcy resistances but increase temperature from Joule heating. Suction produces thinner boundary layers than blowing. Higher thermal relaxation time also reduces the thermal boundary layer thickness and improves the Nusselt number, which improves convective heat transfer. The present findings provide quantitative insight into the regulation of flow and heat transfer in magnetically controlled thermal systems, porous media transport, and advanced cooling processes.</p>