We present a rigorous method of model-order reduction for a class of canonical shear flows, particularly plane Couette flow and pipe flow. In these flows, an extended turbulent state can coexist with the stable laminar. In this case, the boundary between the coexisting basins of attraction, often called the edge of chaos, is the stable manifold of an edge state, a lower-branch traveling wave solution. We show that a low-dimensional submanifold of the edge of chaos can be constructed from velocity data using the recently developed theory of spectral submanifolds (SSMs). These manifolds are the unique smoothest nonlinear continuations of nonresonant spectral subspaces of the linearized system at stationary states. We use very low dimensional SSM-based reduced-order models to predict transitions to turbulence or laminarization for velocity fields near the edge of chaos.