A simple vapor-compression refrigeration system becomes ineffective and inefficient as it consumes a huge energy supply when operating between large temperature differences. Moreover, the recent Kigali amendment has raised a concern about phasing out some hydrofluorocarbon refrigerants due to their impact on the environment. In this paper, a numerical investigation is carried out to compare the performance of a cascade refrigeration system with two environmentally friendly refrigerant combinations, namely, R170–R404A and R41–R404A. Refrigerant R170, from the hydrocarbon category, and refrigerant R41, from the hydrofluorocarbon category, are separately chosen for the low-temperature circuit due to their similar thermophysical properties. On the other hand, refrigerant R404A is selected for the high-temperature circuit. An attempt is made to replace refrigerant R41 with refrigerant R170 as a possible alternative. The condenser temperature is kept constant at 40 °C, and the evaporator temperature is varied from −60 °C to −30 °C. The mathematical model developed for the cascade refrigeration system is solved using Engineering Equation Solver (EES). The effect of evaporator temperature on different performance parameters such as the COP, exergetic efficiency, and total plant cost rate is evaluated. The predicted results show that the thermoeconomic performance of the R170–R404A-based system is marginally lower compared to that of the R41–R404A-based system. The system using refrigerant pair R170–R404A has achieved only a 2.4% lower exergetic efficiency compared to the system using R41–R404A, with an increase in the annual plant cost rate of only USD 200. As the global warming potential (GWP) of R170 is less than that of R41, and R170 belongs to the hydrocarbon category, the use of the R170–R404A combination in a cascade refrigeration system can be recommended as an alternative to R41–R404A.