We investigate the production sites and the enrichment history of $r$-process elements in the Galaxy, as traced by the [Eu/Fe] ratio, using the high resolution, cosmological zoom-in simulation `Eris'. At $z=0$, Eris represents a close analog to the Milky Way, making it the ideal laboratory to understand the chemical evolution of our Galaxy. Eris formally traces the production of oxygen and iron due to Type-Ia and Type-II supernovae. We include in post-processing the production of $r$-process elements from compact binary mergers. Unlike previous studies, we find that the nucleosynthetic products from compact binary mergers can be incorporated into stars of very low metallicity and at early times, even with a minimum delay time of 100 Myr. This conclusion is relatively insensitive to modest variations in the merger rate, minimum delay time, and the delay time distribution. By implementing a first-order prescription for metal-mixing, we can further improve the agreement between our model and the data for the chemical evolution of both [$\alpha$/Fe] and [Eu/Fe].We argue that compact binary mergers could be the dominant source of $r$-process nucleosynthesis in the Galaxy.