We consider the Steiner tree problem in quasi-bipartite graphs, where no two Steiner vertices are connected by an edge. For this class of instances, we present an efficient algorithm to exactly solve the so called directed component relaxation (DCR), a specific form of hypergraphic LP relaxation that was instrumental in the recent break-through result by Byrka et al. [BGRS10] (STOC 2010). Our algorithm hinges on an efficiently computable map from extreme points of the bidirected cut relaxation to feasible solutions of (DCR). As a consequence, together with [BGRS10] we immediately obtain an efficient 73/60-approximation for quasi-bipartite Steiner tree instances. We also present a particularly simple (BCR)-based random sampling algorithm that achieves a performance guarantee slightly better than 77/60.