Link prediction is central to many real-world applications, but its performance may be hampered when the graph of interest is sparse. To alleviate issues caused by sparsity, we investigate a previously overlooked phenomenon: in many cases, a densely connected, complementary graph can be found for the original graph. The denser graph may share nodes with the original graph, which offers a natural bridge for transferring selective, meaningful knowledge. We identify this setting as Graph Intersection-induced Transfer Learning (GITL), which is motivated by practical applications in e-commerce or academic co-authorship predictions. We develop a framework to effectively leverage the structural prior in this setting. We first create an intersection subgraph using the shared nodes between the two graphs, then transfer knowledge from the source-enriched intersection subgraph to the full target graph. In the second step, we consider two approaches: a modified label propagation, and a multi-layer perceptron (MLP) model in a teacher-student regime. Experimental results on proprietary e-commerce datasets and open-source citation graphs show that the proposed workflow outperforms existing transfer learning baselines that do not explicitly utilize the intersection structure.