Language models have achieved impressive performances on dialogue generation tasks. However, when generating responses for a conversation that requires factual knowledge, they are far from perfect, due to an absence of mechanisms to retrieve, encode, and reflect the knowledge in the generated responses. Some knowledge-grounded dialogue generation methods tackle this problem by leveraging facts from Knowledge Graphs (KGs); however, they do not guarantee that the model utilizes a relevant piece of knowledge from the KG. To overcome this limitation, we propose SUbgraph Retrieval-augmented GEneration (SURGE), a framework for generating context-relevant and knowledge-grounded dialogues with the KG. Specifically, our SURGE framework first retrieves the relevant subgraph from the KG, and then enforces consistency across facts by perturbing their word embeddings conditioned by the retrieved subgraph. Then, we utilize contrastive learning to ensure that the generated texts have high similarity to the retrieved subgraphs. We validate our SURGE framework on OpendialKG and KOMODIS datasets, showing that it generates high-quality dialogues that faithfully reflect the knowledge from KG.