Tumor invasion and metastasis are strongly regulated by biophysical interactions between tumor cells and the extracellular matrix (ECM). While the influence of ECM stiffness on cell migration, adhesion, and contractility has been extensively studied in 2D culture, extension of this concept to 3D cultures that more closely resemble tissue has proven challenging, because perturbations that change matrix stiffness often concurrently change cellular confinement. This coupling is particularly problematic given that matrix-imposed steric barriers can regulate invasion speed independent of mechanics. Here we introduce a matrix platform based on microfabrication of channels of defined wall stiffness and geometry that allows independent variation of ECM stiffness and channel width. For a given ECM stiffness, cells confined to narrow channels surprisingly migrate faster than cells in wide channels or on unconstrained 2D surfaces, which we attribute to increased polarization of cell-ECM traction forces. Confinement also enables cells to migrate increasingly rapidly as ECM stiffness rises, in contrast with the biphasic relationship observed on unconfined ECMs. Inhibition of nonmuscle myosin II dissipates this traction polarization and renders the relationship between migration speed and ECM stiffness comparatively insensitive to matrix confinement. We test these hypotheses in silico by devising a multiscale mathematical model that relates cellular force generation to ECM stiffness and geometry, which we show is capable of recapitulating key experimental trends. These studies represent a paradigm for investigating matrix regulation of invasion and demonstrate that matrix confinement alters the relationship between cell migration speed and ECM stiffness.