In the present study, a hydromagnetic non-Newtonian (dilatant) fluid flow in a convergent conduit, in the presence of a variable transverse magnetic field, has been investigated. The governing nonlinear partial differential equations are reduced to system of ordinary differential equations. These equations are solved numerically by the collocation method and implemented in MATLAB. The study determines the flow profiles and the impact of the flow parameters on the flow variables. Joule heating, variable viscosity, viscous dissipation, skin friction, the rate of heat transfer, and the induced magnetic field are taken into account. The obtained results are presented graphically and the impact of varying flow parameters on the skin friction coefficient and the Nusselt number is presented in tabular form. These results indicate that an increase in the Reynolds number, Eckert’s number, and the Joule heating parameter increases the fluid’s velocity, while an increase in the Hartmann number and the unsteadiness parameter decreases the convective heat transfer and the fluid’s velocity. Further, the skin friction coefficient decreases with increase in the Reynolds number, the Hartmann number, and the Joule heating parameter. Therefore, a less viscous fluid is appropriate to facilitate the fluid’s motion, but the presence of high magnetic field reduces the fluid’s motion.