Current theoretical investigation of atherosclerotic arteries deals with mathematical models that represent non-Newtonian flow of blood through a stenosed artery in the presence of a transverse magnetic field. Here, the rheology of the flowing blood is characterised by a generalised Power law model. The distensibility of an arterial wall has been accounted for based on local fluid mechanics. A radial coordinate transformation is initiated to map cosine geometry of the stenosis into a rectangular grid. An appropriate finite difference scheme has been adopted to solve the unsteady non-Newtonian momentum equations in cylindrical coordinate system. Exploiting suitably prescribed conditions based on the assumption of an axial symmetry under laminar flow condition rendered the problem effectively to two dimensions. An extensive quantitative analysis has been performed based on numerical computations in order to estimate the effects of Hartmann number (M), Power law index (n), generalised Reynolds number (Re_G), severity of the stenosis (δ) on various parameters such as flow velocity, flux and wall shear stress by means of their graphical representations so as to validate the applicability of the proposed mathematical model. The present results agree with some of the existing findings in the literature.