Two-dimensional (2D) semiconductors have attracted tremendous interest as natural passivation and atomically thin channels could facilitate continued transistor scaling. However, air-stable 2D semiconductors with high performance are quite elusive. Recently, an extremely-air-stable MoSi2N4 monolayer was successfully fabricated [Hong et al., Science 369, 670 (2020)]. To further reveal its potential application in sub-5-nm metal-oxide-semiconductor field-effect transistors (MOSFETs), there is an urgent need to develop integrated circuits. Here, we report first-principles quantum-transport simulations on the performance limits of n- and p-type sub-5-nm monolayer (ML) MoSi2N4 MOSFETs. We find that the on-state current in the MoSi2N4 MOSFETs can be effectively manipulated by the length of gate and underlap, as well as the doping concentration. Very strikingly, we also find that for the n-type devices the optimized on-state currents can reach up to 1390 and 1025 µA/µm for high-performance and low-power (LP) applications, respectively, both of which satisfy the International Technology Roadmap for Semiconductors (ITRS) requirements. The optimized on-state current can meet the LP application (348 µA/µm) for p-type devices. Finally, we find that the MoSi2N4 MOSFETs have an ultralow subthreshold swing and power-delay product, which have the potential to realize high-speed and low-power consumption devices. Our results show that MoSi2N4 is an ideal 2D channel material for future competitive ultrascaled devices. see https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.16.044022