In the context of algorithm management driven by large language models, reconciling “humanistic attribute” with “technological rigidity” is pivotal for enhancing employee work engagement. Existing studies predominantly focus on the impact of algorithm management on gig workers, yielding inconsistent conclusions. Drawing on the person-environment fit theory, this paper analyzes the potential logic of human-algorithm fit. It explains how algorithm management functionalities (algorithmic decision-making, monitoring, and feedback) and employee participation characteristics (self-efficacy, proactive behavior, and job autonomy) jointly affect work engagement. Using the fuzzy-set Qualitative Comparative Analysis (fsQCA) method, it conducts a configurational analysis of questionnaire survey data from 349 employees. The results show that: (1) No single element is a necessary condition for high work engagement, but job autonomy plays a relatively common role in facilitating such outcomes. (2) Four types of human-algorithm fit configuration generate high work engagement, namely “Supplies-Demands-Abilities” fit, “Needs-Supplies” collaborating with “Demands-Abilities” fit, “Demands-Abilities” collaborating with “Needs” fit, and the “Needs-Demands-Abilities” fit. (3) The Random Forest importance ranking indicates that employees’ intrinsic motivation is more explanatory than technology-driven motivation. This paper confirms the necessity of human-algorithm fit in the organizational context and the complex mechanisms on work engagement, providing insights for organizations to implement algorithm management.