Graphene, a two-dimensional sheet of carbon atoms, is a promising material for next-generation technology because of its advantageous electronic properties, such as extremely high carrier mobilities. However, chemical functionalization schemes are needed to integrate graphene with the diverse range of materials required for device applications. In this paper, we report self-assembled monolayers of the molecular semiconductor perylene-3,4,9,10-tetracarboxylic dianhydride (PTCDA) formed on epitaxial graphene grown on the SiC(0001) surface. The molecules possess long-range order with a herringbone arrangement, as shown by ultra-high vacuum scanning tunnelling microscopy at room temperature. The molecular ordering is unperturbed by defects in the epitaxial graphene or atomic steps in the underlying SiC surface. Scanning tunnelling spectra of the PTCDA monolayer show distinct features that are not observed on pristine graphene. The demonstration of robust, uniform organic functionalization of epitaxial graphene presents opportunities for graphene-based molecular electronics and sensors.