We study the dynamic control of birhythmicity under an impulsive feedback control scheme where the feedback is made ON for a certain rather small period of time and for the rest of the time, it is kept OFF. We show that, depending on the height and width of the feedback pulse, the system can be brought to any of the desired limit cycles of the original birhythmic oscillation. We derive a rigorous analytical condition of controlling birhythmicity using the harmonic decomposition and energy balance methods. The efficacy of the control scheme is investigated through numerical analysis in the parameter space. We demonstrate the robustness of the control scheme in a birhythmic electronic circuit where the presence of noise and parameter fluctuations are inevitable. Finally, we demonstrate the applicability of the control scheme in controlling birhythmicity in diverse engineering and biochemical systems and processes, such as an energy harvesting system, a glycolysis process, and a p53-mdm2 network.