Grid background and dead-time harmonic perturbations pose challenges in maintaining power quality and stability in the grid integration of renewable energy conversion systems. Therefore, a super-twisting algorithm with the ability to reject harmonic perturbations on three-phase grid-tied inverters is proposed. The algorithm design is related to dead-time and grid background harmonics so that robust performance is achieved under these perturbations. To highlight the advantages of the proposed algorithm, it is compared against proportional-integral (PI) control in several dead-time and grid background harmonic sweeps. The maximum inverter current distortion obtained with PI control in simulation is 15.53%. However, by using the proposed algorithm instead, the maximum current distortion is reduced to 1.8%. Dead-time is experimentally swept by using a permanent magnet motor as a harmonic-free grid. Experimental results of grid background harmonics are obtained by injecting reactive energy into a grid with 2.19% voltage distortion. When PI control is used in the experimental setup, the maximum inverter current distortion obtained is 11.4%. However, with the proposed algorithm, the maximum current distortion is reduced to 1.5%, which complies with the standard IEEE 1547-2018. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.