Network alignment, the process of finding correspondences between nodes in different graphs, has significant scientific and industrial applications. We find that many existing network alignment methods fail to achieve accurate alignments because they break up node neighborhoods during alignment, failing to preserve matched neighborhood consistency. To improve this, we propose CONE-Align, which matches nodes based on embeddings that model intra-network proximity and are aligned to be comparable across networks. Experiments on diverse, challenging datasets show that CONE-Align is robust and obtains up to 49% greater accuracy than the state-of-the-art graph alignment algorithms.