We explore the master-slave chaos synchronization of stochastic time-delay Lur'e systems within a networked environment. To tackle the challenges posed by potential mode-mismatch behavior and limited networked channel resources, an asynchronous and adaptive event-triggered (AAET) controller is employed. A criterion on the stochastic stability and ${\mathcal{L}}_2-{\mathcal{L}}_{\mathrm{\infty }}$ disturbance-suppression performance of the synchronization-error system is proposed by using a Lyapunov-Krasovskii functional, a Wirtinger-type inequality, the Itô formula, as well as a convex combination inequality. Then, a method for determining the desired AAET controller gains is proposed by decoupling the nonlinearities that arise from the Lyapunov matrices and controller gains. Finally, the applicability of the AAET control approach is validated by a Chua's circuit.