We present an anisotropic, goal-oriented error estimator based on the Dual Weighted Residual (DWR) framework for nonlinear time-dependent, convection-dominated transport problems. Using elementwise p-anisotropic finite element spaces, the estimator is decomposed direction-wise in space and time, yielding directional error indicators and enabling adaptive anisotropic hp-refinement naturally (anisotropic mesh refinement and elementwise anisotropic p-enrichment). The approach employs discontinuous Galerkin discretizations in space and time, which are particularly well suited for high Peclet-number regimes with sharp internal and boundary layers. We demonstrate efficiency and robustness for multiple target quantities of interest, including representative, relevant goal functionals. The directional indicators quantify goal-relevant anisotropy and drive refinements that efficiently resolve sharp layers while avoiding unnecessary isotropic refinement. Numerical examples for established convection-dominated benchmarks in up to three spatial dimensions confirm the superiority of the proposed anisotropic strategy over standard isotropic h- and hp-adaptive refinement in terms of accuracy-versus-cost and reliable error control.