Biological invasion remains a major threat to biodiversity in general and a disruptor to mutualistic interactions in particular. While a number of empirical studies have directly explored the role of invasion in mutualistic pollination networks, a clear picture is yet to emerge and a theoretical model for comprehension still lacking. Here, using an eco‐evolutionary model of bipartite mutualistic networks with trait‐mediated interactions, we explore invader trait, propagule pressure, and network features of recipient community that contribute importantly to the success and impact of an invasion. High level of invasiveness is observed when invader trait differs from those of the community average, and level of interaction generalization equals to that of the community average. Moreover, multiple introductions of invaders with declining propagules enhance invasiveness. Surprisingly, the most successful invader is not always the one having the biggest impact on the recipient community. The network structure of recipient community, such as nestedness and modularity, is not a primary indicator of its invasibility; rather, the invasibility is best correlated with measurements of network stability such as robustness, resilience, and disruptiveness (a measure of evolutionary instability). Our model encompasses more general scenarios than previously studied in predicting invasion success and impact in mutualistic networks, and our results highlight the need for coupling eco‐evolutionary processes to resolve the invasion dilemma.