Two-dimensional (2D) superconductors are an emerging platform for strongly correlated physics and quantum information science. Their reduced dimensionality, atomically flat interfaces, and high crystallinity are attractive for realizing compact lumped-element devices in superconducting circuits. However, synthesizing large-area, monolayer 2D superconductors remains challenging because of their susceptibility to oxidation. Here, we report an "encapsulation epitaxy" mechanism that enables the growth of large-area, air-stable, monolayer superconducting NbSe2 films and explore their use in superconducting quantum circuits. A 2D encapsulation layer, such as graphene or hexagonal boron nitride (hBN), pre-deposited on a 3D substrate (e.g., SiO2 or Si3N4), serves both as a template for epitaxial growth of monolayer NbSe2 (1L-NbSe2) underneath it and as a protective cover. This approach produces uniform, large-area (>1-inch) 1L-NbSe2 with greatly enhanced ambient stability, enabling device fabrication in air. The resulting 1L-graphene/NbSe2 heterostructures exhibit robust superconductivity (Tc ~ 1 K) and enhanced charge density wave order (TCDW ~ 177 K), indicative of high material quality. We further integrate 1L-NbSe2 into superconducting circuits using oxidation-free transfer and superconducting edge-contact techniques. The 1L-NbSe2 exhibits a measured kinetic inductance LK ~ 0.7 nH/square, making it suitable for quantum circuits requiring high-kinetic-inductance elements. Encapsulation epitaxy thus provides a route to air-stable 2D superconductors and van der Waals heterostructures, with potential for wafer-scale, monolithic fabrication of superconducting quantum circuitry.