The development of artificial intelligence technology has normalized service robots cooperating with front-line employees in restaurants to deliver customer services. The effect of service robots has attracted the attention of several researchers. However, existing research has focused on the relationship between customers and robots, whereas research on employees and robots is limited. This study constructs a conceptual model of the four performance features of service robots and employees’ work well-being from the dual perspectives of innovation resistance theory and conservation of resources theory. Using bootstrapping to test the chain mediation effect of robot anxiety and work engagement, the following conclusions were drawn: (1) the animacy and perceived safety of the service robot have significant indirect effects on employees’ work well-being, and employees’ robot anxiety and work engagement have a significant chain mediated effect on the relationships of “animacy-work well-being” and “perceived safety-work well-being”; (2) the animacy and perceived safety of service robots negatively affect employees’ robot anxiety; (3) employees’ robot anxiety weakens their work engagement and reduces their work well-being; and (4) work engagement can improve employees’ work well-being. This study enriches research on the influence of robots on employees in service enterprises.