Two-dimensional (2-D) materials are of tremendous interest to integrated photonics given their singular optical characteristics spanning light emission, modulation, saturable absorption, and nonlinear optics. To harness their optical properties, these atomically thin materials are usually attached onto prefabricated devices via a transfer process. In this paper, we present a new route for 2-D material integration with planar photonics. Central to this approach is the use of chalcogenide glass, a multifunctional material which can be directly deposited and patterned on a wide variety of 2-D materials and can simultaneously function as the light guiding medium, a gate dielectric, and a passivation layer for 2-D materials. Besides claiming improved fabrication yield and throughput compared to the traditional transfer process, our technique also enables unconventional multilayer device geometries optimally designed for enhancing light-matter interactions in the 2-D layers. Capitalizing on this facile integration method, we demonstrate a series of high-performance glass-on-graphene devices including ultra-broadband on-chip polarizers, energy-efficient thermo-optic switches, as well as graphene-based mid-infrared (mid-IR) waveguide-integrated photodetectors and modulators.