The compact directed acyclic word graph (CDAWG) is the minimal compact automaton that recognizes all the suffixes of a string. Classically the CDAWG has been implemented as an index of the string it recognizes, requiring $o(n)$ space for a copy of the string $T$ being indexed, where $n=|T|$. In this work, we propose using the CDAWG as an index for grammar-compressed strings. While this enables all analyses supported by the CDAWG on any grammar-compressed string, in this work we specifically consider pattern matching. Using the CDAWG index, pattern matching can be performed on any grammar-compressed string in $\mathcal{O}(\text{ra}(m)+\text{occ})$ time while requiring only $\mathcal{O}(\text{er}(T))$ additional space, where $m$ is the length of the pattern, $\text{ra}(m)$ is the grammar random access time, $\text{occ}$ is the number of occurrences of the pattern in $T$, and $\text{er}(T)$ is the number of right-extensions of the maximal repeats in $T$. Our experiments show that even when using a naïve random access algorithm, the CDAWG index achieves state of the art run-time performance for pattern matching on grammar-compressed strings. Additionally, we find that all of the grammars computed for our experiments are smaller than the number of right-extensions in the string they produce and, thus, their CDAWGs are within the best known $\mathcal{O}(\text{er}(T))$ space asymptotic bound.