In this paper, the problem on simultaneous estimation of the actuator and sensor faults is first addressed for a class of Markovian jump systems with mixed mode-dependent time-varying delays. By using a generalized system technique, the original system is first transformed into a descriptor one; its states consist of original states and sensor fault. Then, a Markovian adaptive observer is designed for the descriptor system to provide simultaneous estimations of the state, actuator fault, and sensor fault. In the light of online acquired information, a state-feedback-based fault-tolerant controller is constructed to stabilize the closed-loop system in the presence of the actuator fault. Using the Lyapunov–Krasovskii functions, sufficient and necessity conditions for the existence of designed observer and controller are derived in terms of linear matrix inequalities, which can be solved readily through efficient mathematical tools. Finally, numerical and practical examples are given to validate the effectiveness of the proposed method.