Irreversible visual impairment is often caused by primary angle-closure glaucoma, which could be detected via anterior segment optical coherence tomography (AS-OCT). In this paper, an automated system based on deep learning is presented for angle-closure detection in AS-OCT images. Our system learns a discriminative representation from training data that captures subtle visual cues not modeled by handcrafted features. A multilevel deep network is proposed to formulate this learning, which utilizes three particular AS-OCT regions based on clinical priors: 1) the global anterior segment structure; 2) local iris region; and 3) anterior chamber angle (ACA) patch. In our method, a sliding window-based detector is designed to localize the ACA region, which addresses ACA detection as a regression task. Then, three parallel subnetworks are applied to extract AS-OCT representations for the global image and at clinically relevant local regions. Finally, the extracted deep features of these subnetworks are concatenated into one fully connected layer to predict the angle-closure detection result. In the experiments, our system is shown to surpass previous detection methods and other deep learning systems on two clinical AS-OCT datasets.