The impact of an external magnetic field on the morphology of nanosize clusters obtained by magnetoelectrolysis of CuSO$_4$ aqueous solution within the applied magnetic field up to 0.31 T is investigated by the controlled potential method. As found, the morphology of obtained electrodeposits is significantly changed by the applied magnetic field. In a zero magnetic field, one has an isotropic growth of deposited aggregates, whereas for nonzero fields, there appear preferred directions of growth for the field oriented either perpendicular or parallel to the electric current lines. The observed ramified electrodeposits have the fractal structure, with the fractal dimension being dependent on the magnetic field strength and orientation. The features observed can be explained taking into account the magnetohydrodynamic (MHD) convection induced by the Lorentz force, which affects the natural convection due to concentration gradient in electrolyte near the cathode surface both along the plane of the electrode and perpendicular to it. In this way, nanoobjects of various structures (magnetic-field-driven) and different size (electrolysis-time-dependent) can be fabricated in a controlled way. Additional feature observed in the present work, viz., change of the sign of the potential difference between the lower and upper probes with increase of the magnetic field, evidences the corresponding change of sign of the concentration gradient along the cathode surface. This feature allows us to explain the impact of magnetic field in the parallel orientation (where no Lorentz forces act) by interplay of the MHD effects and the diffusion over the cathode plane.