Accurate and autonomous transportation of micro-objects and biological cells can enable significant advances in a wide variety of research disciplines. Here, we present a novel, vision-based, model-free microrobotic pushing algorithm for the autonomous manipulation of micro objects and biological cells. The algorithm adjusts the axis of a rotating magnetic field that in turn controls the heading angle and spin axis of a spherical Janus rolling microrobot. We introduce the concept of a microrobotic guiding corridor to constrain the object and to avoid pushing failures. We then show that employing only two simple conditions, the microrobot is able to successfully and autonomously push microscale objects along predefined trajectories. We evaluate the performance of the algorithm by measuring the mean absolute error and completion time relative to a desired path at different actuation frequencies and guiding corridor widths. Finally, we demonstrate biomedical applicability by autonomously transporting a single biological cell, highlighting the methods potential for applications in tissue engineering, drug delivery and synthetic biology.