This paper proposes a novel nonfragile robust asynchronous control scheme for master-slave uncertain chaotic Lurie network systems with randomly occurring time-varying parameter uncertainties and controller gain fluctuation. The asynchronous phenomenon occurs between the system modes and the controller modes. In order to consider a more realistic situation in designing a reliable proportional-derivative controller, Bernoulli stochastic process and memory feedback are introduced to the concept of nonlinear control system. First, by taking full advantage of the additional derivative state term and variable multiple integral terms, a newly augmented Lyapunov-Krasovskii functional is constructed via an adjustable parameter. Second, based on new integral inequalities including almost all of the existing integral inequalities, which can produce more accurate bounds with more orthogonal polynomials considered, less conservative synchronization criteria are obtained. Third, a desired nonfragile estimator controller is achieved under the aforementioned methods. Finally, 4 numerical simulation examples of Chua's circuit and 3-cell cellular neural network with multiscroll chaotic attractors are presented to illustrate the effectiveness and advantages of the proposed theoretical results.