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Delay is an inherent characteristic of many industrial systems, which can lead to the decline of system control performance and even affect the stability of the system. In practical systems, the characteristics of finite-time systems deserve more attention. In view of the above situation, the design of finite-time controller for a class of Markov jump systems with time-delay is studied. By relaxing the condition that the transition probability is fully known to a more general case where the transition probability is partially unknown, the free weight method is adopted to ensure that the obtained linear matrix inequalities are less conservative. Firstly, the criteria of finite-time boundedness and finite-time Boundedness for Markov jumping systems are given. Then, the gain matrix of the state observer and the state feedback controller is obtained by solving the linear matrix inequality (LMIs). Finally, a simulation example is given to verify the effectiveness of the proposed algorithm.