This paper addresses the control problem of a class of complex dynamical networks with each node being a Lur’e system whose nonlinearity satisfies a sector condition, by applying local feedback injections to a small fraction of the nodes. The pinning control problem is reformulated in the framework of the absolute stability theory. It is shown that the global stability of the controlled network can be reduced to the test of a set of linear matrix inequalities, which in t guarantee the absolute stability of the corresponding Lur’e systems whose dimensions are the same as that of a single node. A circle-type criterion in the frequency domain is further presented for checking the stability of the controlled network graphically. Finally, a network of Chua’s oscillators is provided as a simulation example to illustrate the effectiveness of the theoretical results.