Optical Coherence Tomography (OCT) is an imaging modality that has been widely adopted for visualizing corneal, retinal and limbal tissue structure with micron resolution. It can be used to diagnose pathological conditions of the eye, and for developing pre-operative surgical plans. In contrast to the posterior retina, imaging the anterior tissue structures, such as the limbus and cornea, results in B-scans that exhibit increased speckle noise patterns and imaging artifacts. These artifacts, such as shadowing and specularity, pose a challenge during the analysis of the acquired volumes as they substantially obfuscate the location of tissue interfaces. To deal with the artifacts and speckle noise patterns and accurately segment the shallowest tissue interface, we propose a cascaded neural network framework, which comprises of a conditional Generative Adversarial Network (cGAN) and a Tissue Interface Segmentation Network (TISN). The cGAN pre-segments OCT B-scans by removing undesired specular artifacts and speckle noise patterns just above the shallowest tissue interface, and the TISN combines the original OCT image with the pre-segmentation to segment the shallowest interface. We show the applicability of the cascaded framework to corneal datasets, demonstrate that it precisely segments the shallowest corneal interface, and also show its generalization capacity to limbal datasets. We also propose a hybrid framework, wherein the cGAN pre-segmentation is passed to a traditional image analysis-based segmentation algorithm, and describe the improved segmentation performance. To the best of our knowledge, this is the first approach to remove severe specular artifacts and speckle noise patterns (prior to the shallowest interface) that affects the interpretation of anterior segment OCT datasets, thereby resulting in the accurate segmentation of the shallowest tissue interface. To the best of our knowledge, this is the first work to show the potential of incorporating a cGAN into larger deep learning frameworks for improved corneal and limbal OCT image segmentation. Our cGAN design directly improves the visualization of corneal and limbal OCT images from OCT scanners, and improves the performance of current OCT segmentation algorithms.