The high carrier mobility, appropriate band gap and good environmental stability of two-dimensional (2D) MoSi2N4 enable it to be an appropriate channel material for transistors with excellent performance. Therefore, we predict the performance of double-gate (DG) metal-oxide-semiconductor field-effect transistors (MOSFETs) based on monolayer (ML) MoSi2N4 by ab initio quantum-transport calculations. The results show that the on-state current of the p-type device is remarkable when the gate length is greater than 4 nm, which can meet the high performance requirements of the International Technology Roadmap for Semiconductors (ITRS), 2013 version. Moreover, the gate length can be reduced to 3 nm when an underlap (UL) structure is employed in the MOSFET, and the sub-threshold swing, intrinsic delay time and power consumption also perform well. The calculation results reveal that ML MoSi2N4 will be a promising alternative for transistor channel materials in the post-silicon era.