We develop a set of tools for the asymptotic analysis of minimizers of the anisotropic Ginzburg-Landau energy functional among the admissible competitors with Dirichlet boundary datum of negative degree $-D$. As a byproduct of our analysis, we prove that the energy of a minimizer is $K\ln (1/\varepsilon)+o(\ln (1/\varepsilon))$, where $K$ depends only on $D$ and the material constants that enter into the expression for the energy.