One of the challenges when simulating astrophysical flows with self-gravity is to compute thegravitational forces. In contrast to the hyperbolic hydrodynamic equations, the gravity field isdescribed by an elliptic Poisson equation. We present a purely hyperbolic approach by reformulatingthe elliptic problem into a hyperbolic diffusion problem, which is solved in pseudotime, using the same explicit high-order discontinuous Galerkin method we use for the flow solution. Theflow and the gravity solvers operate on a joint hierarchical Cartesian mesh and are two-way coupledvia the source terms. A key benefit of our approach is that it allows the reuse of existingexplicit hyperbolic solvers without modifications, while retaining their advanced features such asnon-conforming and solution-adaptive grids. By updating the gravitational field in each Runge-Kuttastage of the hydrodynamics solver, high-order convergence is achieved even in coupled multi-physicssimulations. After verifying the expected order of convergence for single-physics and multi-physicssetups, we validate our approach by a simulation of the Jeans gravitational instability.Furthermore, we demonstrate the full capabilities of our numerical framework by computing aself-gravitating Sedov blast with shock capturing in the flow solver and adaptive mesh refinementfor the entire coupled system.