This paper presents a novel application of the Fractal Fractional derivative to control the flow of non-Newtonian fluids. The study focuses on the generalized magnetohydrodynamic (MHD) flow of a Casson-type hybrid nanofluid in a micro-channel and its modeling and solution. The hybrid nanofluid mixed with the suspension of copper (Cu) and titanium dioxide (TiO2) nanoparticles with carboxymethyl cellulose (CMC) and water (H2O) as base fluids are considered. First, the partial differential equations are converted to a fractional model with the current and new description of fractional derivatives, i.e., Fractal fractional derivatives. Then, ordinary differential equations are resolved numerically using a fractional Laplace transformation. Finally, the results are numerically calculated and shown in various graphs with a physical description to study the physical importance of different relevant factors. As a result, it's computed that the momentum field indicates the decaying behavior with incrementing the Casson fluid parameter. Furthermore, the impact of CMC-based hybrid nanofluid (CMC + Ag + TiO2) on energy and momentum is slightly more significant than the water-based (H2O + Ag + TiO2) hybrid nanofluid. Moreover, due to the impact of volume fraction, the fluid temperature increases while velocity slows down due to increased thermal conductivity.