Aharonov-Bohm oscillations effectively demonstrate coherent, ballistic transport in mesoscopic rings and tubes. In three-dimensional topological insulator nanowires, they can be used to not only characterize surface states but also to test predictions of unique topological behaviour. Here we report measurements of Aharonov-Bohm oscillations in (Bi1.33Sb0.67)Se3 that demonstrate salient features of topological nanowires. By fabricating quasi-ballistic three-dimensional topological insulator nanowire devices that are gate-tunable through the Dirac point, we are able to observe alternations of conductance maxima and minima with gate voltage. Near the Dirac point, we observe conductance minima for zero magnetic flux through the nanowire and corresponding maxima (having magnitudes of almost a conductance quantum) at magnetic flux equal to half a flux quantum; this is consistent with the presence of a low-energy topological mode. The observation of this mode is a necessary step towards utilizing topological properties at the nanoscale in post-CMOS applications.