This work presents an analytical study of unsteady, one-dimensional magnetohydrodynamic flow of a Casson–Brinkman fluid over an infinite vertical plate, incorporating heat and mass transfer, internal heat generation, and a first-order chemical reaction. The plate velocity, temperature, and concentration are time-dependent, with ramped boundary conditions, and the governing equations account for a transverse magnetic field. Using Buckingham’s π-theorem, the model is nondimensionalized, introducing key parameters including the Grashof numbers, Hartmann number, Prandtl number, Schmidt number, Casson parameter, and Brinkman parameter. The classical Fourier and Fick laws are extended using the Caputo fractional derivative to capture memory effects, yielding a time-fractional model. The coupled fractional partial differential equations are solved analytically via Laplace transforms, and the effects of the fractional order and the physical parameters on the velocity, temperature, and concentration profiles are graphically analyzed. Results reveal that the fractional parameter significantly varies the heat and mass transfer profiles.