Accurate and repeatable delineation of corneal tissue interfaces is necessary for surgical planning during anterior segment interventions, such as Keratoplasty. Designing an approach to identify interfaces, which generalizes to datasets acquired from different Optical Coherence Tomographic (OCT) scanners, is paramount. In this paper, we present a Convolutional Neural Network (CNN) based framework called CorNet that can accurately segment three corneal interfaces across datasets obtained with different scan settings from different OCT scanners. Extensive validation of the approach was conducted across all imaged datasets. To the best of our knowledge, this is the first deep learning based approach to segment both anterior and posterior corneal tissue interfaces. Our errors are 2x lower than non-proprietary state-of-the-art corneal tissue interface segmentation algorithms, which include image analysis-based and deep learning approaches.