The past decade has witnessed the burgeoning discovery of a variety of topological states of matter with distinct nontrivial band topologies. Thus far, most of materials studied possess two-dimensional or three-dimensional electronic structures, with only a few exceptions that host quasi-one-dimensional (quasi-1D) topological electronic properties. Here we present the clear-cut evidence for Dirac fermions in the quasi-1D telluride TaNiTe5. We show that its transport behaviors are highly anisotropic and we observe nontrivial Berry phases via the quantum oscillation measurements. The nontrivial band topology is further corroborated by first-principles calculations. Our results may help to guide the future quest for topological states in this new family of quasi-1D ternary chalcogenides.