In this article we construct a combinatorial quasi-free differential graded model for the Orlik-Solomon algebra of supersolvable matroids, which generalizes in a matroidal setting the cdga of admissible graphs introduced by M. Kontsevich for the braid arrangements. Our construction draws on well-known concepts from matroid theory, including modularity, single-element extensions, and generalized parallel connections. We also show that this model carries a cooperadic structure in a suitably generalized sense. As an application, we use this model to give a new proof that the Orlik-Solomon algebras of supersolvable matroids are Koszul.