This paper aims to investigate the finite-time control problem of time-delay switched positive systems under asynchronous switching. That is, for the asynchronous phenomenon caused by the switching of the controller lagging behind the switching of the subsystems, a finite-time stabilization research is carried out for switched positive systems based on the average dwell time switching method. Firstly, the running interval of each positive subsystem is divided into controller matched interval and controller mismatched interval, and the multiple co-positive Lyapunov-Krasovskii functional is constructed. Then, based on the finite-time stability theory, the suitable average dwell time switching law and controllers that guarantee the time-delay switched positive system under asynchronous switching finite-time stable are designed, and sufficient conditions of the finite-time stabilization for both continuous time and discrete time systems are given. Finally, two examples are provided to verify the effectiveness of the proposed method.