The multi-stage continuous stirred tank reactor (CSTR) system is widely used in chemical processes, yet its multivariable coupling and complex nonlinear dynamics pose significant challenges for control. To address this, a modeling approach based on the deep Koopman operator and a model predictive control (MPC) approach with a three-term objective function are proposed. The deep Koopman operator is employed to map the dynamics of the multi-stage CSTR system into a high-dimensional linear space, within which the MPC algorithm is designed to enhance the control performance. Simulation results demonstrate that the deep Koopman model exhibits high accuracy in multi-step prediction tasks, with the relative mean error of both concentration and temperature states maintained below 0.10%. Compared with the traditional Koopman MPC based on extended dynamic mode decomposition (EDMD), the proposed method significantly reduces the root mean square error and achieves a much lower average computation time than the nonlinear MPC algorithm. Moreover, by incorporating a state increment penalty term, the proposed controller effectively suppresses overshoot while maintaining a comparable response speed to the other two MPC strategies.