Large Language Models (LLMs) can generate high-quality arguments, yet their ability to engage in nuanced and persuasive communicative actions remains largely unexplored. This work explores the persuasive potential of LLMs through the framework of Jürgen Habermas' Theory of Communicative Action. It examines whether LLMs express illocutionary intent (i.e., pragmatic functions of language such as conveying knowledge, building trust, or signaling similarity) in ways that are comparable to human communication. We simulate online discussions between opinion holders and LLMs using conversations from the persuasive subreddit ChangeMyView. We then compare the likelihood of illocutionary intents in human-written and LLM-generated counter-arguments, specifically those that successfully changed the original poster's view. We find that all three LLMs effectively convey illocutionary intent -- often more so than humans -- potentially increasing their anthropomorphism. Further, LLMs craft sycophantic responses that closely align with the opinion holder's intent, a strategy strongly associated with opinion change. Finally, crowd-sourced workers find LLM-generated counter-arguments more agreeable and consistently prefer them over human-written ones. These findings suggest that LLMs' persuasive power extends beyond merely generating high-quality arguments. On the contrary, training LLMs with human preferences effectively tunes them to mirror human communication patterns, particularly nuanced communicative actions, potentially increasing individuals' susceptibility to their influence.