Affected by the COVID-19 pandemic and the international development pattern, the international environment has undergone profound changes. Enterprises, as the main body of activities on the front line of production and operation and the main battlefield of market competition, are facing various risk challenges. In both domestic and international markets, these challenges are becoming increasingly complex for businesses to navigate. For theoretical research, the impact of organizational change on employee innovation performance has become a key issue in organizational behavior and human resource management research. However, the influence mechanism of organizational change on employee innovation performance is still unclear. In this study, we examine whether, how, and when organizational change increases employee innovation performance in accordance with job demands-resource theory, as well as the effect of work pressure and work engagement on employee innovation performance. Data from 289 employees at three time points are examined. The results show that: (1) Organizational change negatively affects employee innovation performance through work pressure, i.e., work pressure mediates the impact of organizational change on employee innovation performance. (2) Organizational change positively affects employee innovation performance through work engagement, i.e., work engagement mediates the impact of organizational change on employee innovation performance. (3) Organizational identity plays a moderating role between organizational change and work pressure and work engagement, respectively, and there is a moderating effect in the process of mediation of work pressure and work engagement. The findings of this study provide important insights into how and when organizational change influences employee innovation performance.