The propagation of cracks driven by a pressurized fluid emerges in several areas of engineering, including structural, geotechnical, and petroleum engineering. In this paper, we present a robust numerical framework to simulate fluid-driven fracture propagation that addresses the challenges emerging in the simulation of this complex coupled nonlinear hydro-mechanical response. We observe that the numerical difficulties stem from the strong nonlinearities present in the fluid equations as well as those associated with crack propagation, from the quasi-static nature of the problem, and from the a priori unknown and potentially intricate crack geometries that may arise. An additional challenge is the need for large scale simulation owing to the mesh resolution requirements and the expected 3D character of the problem in practical applications. To address these challenges we model crack propagation with a high-order hybrid discontinuous Galerkin/cohesive zone model framework, which has proven massive scalability properties, and we model the lubrication flow inside the propagating cracks using continuous finite elements, furnishing a fully-coupled discretization of the solid and fluid equations. We find that a conventional Newton–Raphson solution algorithm is robust even in the presence of crack propagation. The parallel approach for solving the linearized coupled problem consists of standard iterative solvers based on domain decomposition. The resulting computational approach provides the ability to conduct highly-resolved and quasi-static simulations of fluid-driven fracture propagation with unspecified crack path. We conduct a series of numerical tests to verify the computational framework against known analytical solutions in the toughness and viscosity dominated regimes and we demonstrate its performance in terms of robustness and parallel scalability, enabling simulations of several million degrees of freedom on hundreds of processors.