Dialogue act (DA) classification has been studied for the past two decades and has several key applications such as workflow automation and conversation analytics. Researchers have used, to address this problem, various traditional machine learning models, and more recently deep neural network models such as hierarchical convolutional neural networks (CNNs) and long short-term memory (LSTM) networks. In this paper, we introduce a new model architecture, directed-acyclic-graph LSTM (DAG-LSTM) for DA classification. A DAG-LSTM exploits the turn-taking structure naturally present in a multi-party conversation, and encodes this relation in its model structure. Using the STAC corpus, we show that the proposed method performs roughly 0.8% better in accuracy and 1.2% better in macro-F1 score when compared to existing methods. The proposed method is generic and not limited to conversation applications.