While there are some differences, the two concepts are closely related.
Both flow and work engagement are "positive emotional states" (Schaufeli, Bakker and Van Rhenen, 2009, p. 898).
According to Schaufeli et al. (2009, p. 895) "recent research suggests that vigor and dedication constitute the core of engagement, whereas absorption seems to be related to the concept of flow" as defined by Csikszentmihalyi and LeFevre (1989).
"However, typically, flow is a more complex concept that includes many aspects and refers to rather particular, short-term ‘peak’ experiences instead of a more pervasive and persistent state of mind, as is the case with engagement" (Schaufeli et al., 2002, p. 75). Moreover, work engagement refers specifically to "identification with one's work" (Schaufeli et al., 2009, p. 895).
Csikszentmihalyi, M., & LeFevre, J. (1989). Optimal experience in work and leisure. Journal of Personality and Social Psychology, 56(5), 815–822. https://doi.org/10.1037/0022-35184.108.40.2065
Schaufeli, W. B., Bakker, A. B., & Van Rhenen, W. (2009). How changes in job demands and resources predict burnout, work engagement, and sickness absenteeism. Journal of Organizational Behavior, 30(7), 893–917. https://doi.org/10.1002/job.595
Schaufeli, W., Salanova, M., González-romá, V., & Bakker, A. (2002). The Measurement of Engagement and Burnout: A Two Sample Confirmatory Factor Analytic Approach. Journal of Happiness Studies, 3(1), 71–92. https://doi.org/10.1023/A:1015630930326