In this work, we introduced the concept of a timelike acceleration-slant curve in three-dimensional Minkowski space, characterized by the property that the scalar product of a fixed unit direction and its acceleration vector remained constant. Moreover, we provided a classification of timelike acceleration-slant curves with a non-null axis based on their curvature properties. Later on, we explored the relationships between timelike acceleration-slant curves and various notable curves, including timelike helices, timelike slant helices, timelike Lorentzian spherical curve, and timelike Salkowski curves. To further illustrate and validate the theoretical results, we utilized Mathematica to generate visual representations.