Previous studies on cross-domain sentiment classification depend on the pivot features or utilize the target data for representation learning, which ignore the semantic relevance between different domains. To this end, we exploit Abstract Meaning Representation (AMR) to help with cross-domain sentiment classification. Compared with the textual input, AMR reduces data sparsity and explicitly provides core semantic knowledge and correlations between different domains. In particular, we develop an algorithm to construct a sentiment-driven semantic graph from sentence-level AMRs. We further design two strategies to linearize the semantic graph and propose a text-graph interaction model to fuse the text and semantic graph representations for cross-domain sentiment classification. Empirical studies show the effectiveness of our proposed model over several strong baselines. The results also indicate the importance of the proposed sentiment-driven semantic graph for cross-domain sentiment classification.