Illusory pattern perception is a well-documented human cognitive tendency to infer meaningful relationships in data that is actually random. Such a tendency, often described as "connecting the dots" where none exist, can result in systematic reasoning errors. This paper investigates whether Large Language Models (LLMs) exhibit such perceptual tendencies, which can lead to systematic errors in downstream applications. To our knowledge, this work presents the first systematic study of illusory pattern perception in LLMs, adapting classic psychological paradigms to three tasks with direct empirical comparison to human behaviors. We find that LLMs frequently exhibit stronger illusory pattern perception than humans. In particular, models tend to over-associate frequent positive attributes with majority groups or large organizations, and show increased tendencies to construct causal narratives from ambiguous events. To uncover the mechanism behind these behaviors, we develop a feature interpretability framework based on Sparse Autoencoders (SAEs) to analyze internal representations. Our results reveal that holistic frequency perception and analytic cognitive orientation are linked to the emergence of illusory perceptions. These findings highlight a previously underexplored cognitive-like illusion that may affect the reliability of LLM reasoning. Code available at https://github.com/NusIoraPrivacy/illusory.