Topology and disorder have a rich combined influence on quantum transport. In order to probe their interplay, we synthesized one-dimensional chiral symmetric wires with controllable disorder via spectroscopic Hamiltonian engineering, based on the laser-driven coupling of discrete momentum states of ultracold atoms. Measuring the bulk evolution of a topological indicator following a sudden quench, we observed the topological Anderson insulator phase, in which added disorder drives the band structure of a wire from topologically trivial to non-trivial. In addition, we observed the robustness of topologically non-trivial wires to weak disorder and measured the transition to a trivial phase in the presence of strong disorder. Atomic interactions in this quantum simulation platform may enable realizations of strongly interacting topological fluids.