Abstract Matching games form a class of coalitional games that attracted much attention in the literature. Indeed, several results are known about the complexity of computing over them solution concepts. In particular, it is known that computing the Shapley value is intractable in general, formally #P-hard, and feasible in polynomial time over games defined on trees. In fact, it was an open problem whether or not this tractability result holds over classes of graphs properly including acyclic ones. The main contribution of the paper is to provide a positive answer to this question, by showing that the Shapley value is tractable for matching games defined over graphs having bounded treewidth. The proposed polynomial-time algorithm has been implemented and tested on classes of graphs having different sizes and treewidth up to three