This paper addresses the finite-time H∞ control problem of fuzzy semi-Markovian jump systems with immeasurable premise variables by a new observer-based adaptive event-triggered scheme. More general nonlinear systems can be described by the fuzzy system model since the selected premise variables are dependent on system states. Firstly, our work relaxes the restriction of measurable premise variables of nonlinear plant and the fuzzy observer is designed to estimate unmeasurable system states. Then, our work proposes the new observer-based adaptive event-triggered scheme to better save network resources, which is verified by sufficient comparison experiments in the simulation. Furthermore, a novel method is firstly proposed to deal with the asynchronous problem of premise variables, i.e. the premise variables of nonlinear plant, fuzzy observer and event-triggered controller are asynchronous. By applying the delay system method, free-weighting matrix approach and slack matrix variables, novel sufficient conditions are established to guarantee the finite-time H∞ performance of the closed-loop system, where the nonlinear terms caused by network-induced delay and partly unknown transition rates can be tackled by suitable method. Finally, the single-link robot arm system is presented to demonstrate the efficiency and advantages of the proposed fuzzy control scheme.