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This study examines theoretically and computationally the transient magneto-hydrodynamic boundary layer flow and heat transfer in an incompressible rotating nanofluid over a stretching continuous sheet, with a transverse magnetic field applied normal to the sheet plane. The three-dimensional conservation equations for mass, momentum, energy and species (nanoparticle) diffusion, are normalized into a system of two-dimensional dimensionless boundary layer equations, using appropriate scaling transformations. The resulting nanofluid transport model incorporates a Brownian motion parameter, thermophoresis parameter, rotation parameter, unsteady parameter, Prandtl number, Hartmann magnetic parameter and Lewis number, and physically realistic boundary conditions at the sheet surface and in the free stream. The nonlinear two-point boundary value problem is solved using a robust, efficient finite element method based on the variational formulation. A detailed evaluation of the effects of the governing physical parameters on the velocity components, temperature and nanoparticle concentration via graphical plots is conducted. Primary velocity is strongly retarded with increasing Hartmann number and there is also a reduction in secondary velocity magnitude. Both temperature and nanoparticle concentration are positively affected by the Hartmann number. Increasing rotational parameter decreases both primary and secondary velocity, and also depresses temperature and nanoparticle concentration. Unsteadiness parameter is generally found to enhance primary velocity and temperatures but exhibits a varied influence on secondary velocity and nanoparticle concentration. The reduced Nusselt number (wall temperature gradient) is observed to be depressed with both Brownian motion and thermophoresis effects, whereas the contrary behaviour is computed for the reduced Sherwood number (wall mass transfer gradient). The reduced Nusselt number and the Sherwood number also show a steady decrease with increasing rotational parameter. The present finite element method solutions have been validated extensively with the previously published results, demonstrating excellent correlation. The study has important applications in the manufacture and electromagnetic control of complex magnetic nanofluid materials of relevance to biomedical, energy systems and aerospace systems technologies.