This work describes the first cell-based model of tumor-induced angiogenesis. At the extracellular level, the model describes diffusion, uptake, and decay of tumor-secreted pro-angiogenic factor. At the cellular level, the model uses the cellular Potts model based on system-energy reduction to describe endothelial cell migration, growth, division, cellular adhesion, and the evolving structure of the stroma. Numerical simulations show: 1), different tumor-secreted pro-angiogenic factor gradient profiles dramatically affect capillary sprout morphology; 2), average sprout extension speeds depend on the proximity of the proliferating region to the sprout tip, and the coordination of cellular functions; and 3), inhomogeneities in the extravascular tissue lead to sprout branching and anastomosis, phenomena that emerge without any prescribed rules. This model provides a quantitative framework to test hypotheses on the biochemical and biomechanical mechanisms that control tumor-induced angiogenesis.