Marburg virus disease poses a significant risk to global health, impacting both humans and non-human primates. This study has yielded an optimal control model for potentially mitigating the transmission of the Marburg infection. The proposed mathematical model includes fractional-order derivatives in the Caputo sense. Initially, we analyzed the model without control measures, examining its key characteristics regarding local and global stabilities. Subsequently, we extended the model by incorporating suitable time-dependent optimal control variables. We have also introduced two time-dependent control measures: ${\mathrm{\Psi }}_1$ for the prevention of human-to-human Marburg transmission, and ${\mathrm{\Psi }}_2$ to enhance the rate of quarantine of exposed individuals. We performed simulation analysis for both cases i.e., with and without optimal controls using the two-step Newton polynomial approximation method, considering both fractional and classical orders. The numerical findings of the comparative study between classical and fractional cases validate the biological significance of the fractional operator and effectiveness of the proposed optimal control strategies.