This research deals with the problem of finding a minimum energy connected structure in wireless sensor networks that guarantees total target coverage. The problem is formulated as a Node Weighted Steiner Tree and an integer model to solve it is proposed. In order to solve the problem, a branch-and-cut algorithm based on Benders' decomposition is provided. Additionally, a more general algorithm is provided by considering an exponential set of constraints used to guarantee connectivity. An extensive set of experiments is used to demonstrate the effectiveness of the proposed approaches. The results are promising and indicate that the methods are efficient to solve moderated size instances in low computational times.