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This paper investigates global synchronization of asymmetrically coupled dynamical networks with nonidentical nodes in the sense of boundedness. A novel bounded synchronization criterion is presented by checking the validity of inequality involving the second smallest eigenvalue of a redefined symmetric matrix associated with the asymmetric Laplacian matrix. In particular, this criterion can be used to determine the global exponential synchronization of asymmetrically coupled networks with identical nodes by the proposed symmetrization operation, without assuming the symmetry or irreducibility of the coupling matrix. Comparing with the existing contributions, our synchronization result is less conservative and can overcome the complexity of calculating eigenvalues of an asymmetric Laplacian matrix. Numerical experiments are carried out to demonstrate the effectiveness of the method. © 2014 Elsevier B.V.