In this study, we explored the nomological network of team work engagement - a shared and emergent work-related state of well-being composed of team vigor, team dedication, and team absorption. Building on team effectiveness theory and the IMOI model, we proposed that team social exchange is positively related to team work engagement which then predicts supervisor-rated team innovative work behaviors and team in- role work performance. Moreover, we hypothesized that task interdependence strengthens the relationship between team social exchange and team work engagement, and that outcome interdependence strengthens the relationship between team work engagement and team effectiveness. Using a longitudinal, lagged and supervisor-rated data in 72 teams, our hypotheses were supported, lending clear support to the team work engagement construct. Hence, our findings have crucial implications for team work engagement, team effectiveness theory and the importance of team context. Given the increasing predominance of teams in work settings, the findings of this study also have important practical implications for organizations to foster engaged teams.