Dialogue system is designed to converse with humans in a natural way. As an essential part of dialogue system, dialogue generation aims to generate proper response given historical context. Recently, sequence-to-sequence (seq2seq) based models have achieved great success but suffer from ungrammatical problems. In this paper, we propose a Syntax-aware Dialogue Generation (SynDG) model that incorporates syntactic information to generate grammatical responses with an encoder-decoder framework. Specifically, we first construct a syntax-graph with a dependency parser on the dialogue corpus. Then, we employ three graph embedding algorithms to learn syntactic word representations as the input of seq2seq framework. Furthermore, we devise training strategies to predict syntactic structure of the sentence for sufficient syntax understanding. Our empirical study on two multi-turn dialogue datasets demonstrates the effectiveness of SynDG in generating natural and grammatical responses