Sarcasm is widely utilized on social media platforms such as Twitter and Reddit. Sarcasm detection is required for analyzing people's true feelings since sarcasm is commonly used to portray a reversed emotion opposing the literal meaning. The syntactic structure is the key to make better use of commonsense when detecting sarcasm. However, it is extremely challenging to effectively and explicitly explore the information implied in syntactic structure and commonsense simultaneously. In this paper, we apply the pre-trained COMET model to generate relevant commonsense knowledge, and explore a novel scenario of constructing a commonsense-augmented sentiment graph and a commonsense-replaced dependency graph for each text. Based on this, a Commonsense Sentiment Dependency Graph Convolutional Network (CSDGCN) framework is proposed to explicitly depict the role of external commonsense and inconsistent expressions over the context for sarcasm detection by interactively modeling the sentiment and dependency information. Experimental results on several benchmark datasets reveal that our proposed method beats the state-of-the-art methods in sarcasm detection, and has a stronger interpretability.