Weighted graph matching is one of the most fundamental graph theoretical problems used in network design, especially in wireless network services such as radio resource allocation, physical layer security, energy-efficient partner selection and optimizing storage capacity. In this paper we present a distributed heuristic which provides nearly-optimal weighted matchings in polynomial time. We also propose an algorithm to decrease the number of transmitted messages to provide energy-efficient operation. Our approaches assume the asynchronous communication model and small messages having O(log(n)) bits. These assumptions directly fit the battery powered wireless networks such as wireless ad hoc and sensor networks. To the best of our knowledge, we propose the first distributed weighted matching algorithm which benefits from augmentation of augmenting paths having size larger than 3 for these networks. We also provide results of our simulations on synthetic geometric graphs to model wireless networks. Extensive simulations reveal that our algorithm improves the approximation performances of the other weighted matching algorithms and closes the gap between the approximation ratio and the optimum up to 32%. (C) 2018 Elsevier Ltd. All rights reserved.