The supply chain is an extremely important and complex system, which has extensive connections with management and social systems. Since the supply chain system is characterized by non-instantaneous impulses of sudden increase or decrease in demand over a period of time, it is sensitive to various uncertainties and external disturbances, and generates chaotic behavior. Therefore, a supply chain model with non-instantaneous impulses is established, and the effects of non-instantaneous impulses on stable and chaotic supply chain systems are analyzed respectively. At the same time, the influence of the combined effect of distribution and production on the model is discussed, and the finite-time stability strategy is given. In addition, synchronization control is an important method to reduce the negative impact of uncertainty and disturbance on the supply chain. This paper also discusses the non-instantaneous impulsive synchronization behavior of two supply chain systems with the same structure, and gives the corresponding finite-time synchronization strategy. Finally, numerical examples are given to verify the effectiveness of the obtained results. It is found that the non-instantaneous impulse has a significant impact on the supply chain system. But the finite-time stability and synchronization control of the supply chain system can be realized as long as the appropriate non-instantaneous impulsive time interval and intensity are selected.