Spoken dialogue systems typically use a list of top-N ASR hypotheses for inferring the semantic meaning and tracking the state of the dialogue. However ASR graphs, such as confusion networks (confnets), provide a compact representation of a richer hypothesis space than a top-N ASR list. In this paper, we study the benefits of using confusion networks with a state-of-the-art neural dialogue state tracker (DST). We encode the 2-dimensional confnet into a 1-dimensional sequence of embeddings using an attentional confusion network encoder which can be used with any DST system. Our confnet encoder is plugged into the state-of-the-art Global-locally Self-Attentive Dialogue State Tacker (GLAD) model for DST and obtains significant improvements in both accuracy and inference time compared to using top-N ASR hypotheses.