Objective: Glaucoma is the second leading cause of blindness worldwide. Glaucomatous progression can be easily monitored by analyzing the degeneration of retinal ganglion cells (RGCs). Many researchers have screened glaucoma by measuring cup-to-disc ratios from fundus and optical coherence tomography scans. However, this paper presents a novel strategy that pays attention to the RGC atrophy for screening glaucomatous pathologies and grading their severity. Methods: The proposed framework encompasses a hybrid convolutional network that extracts the retinal nerve fiber layer, ganglion cell with the inner plexiform layer and ganglion cell complex regions, allowing thus a quantitative screening of glaucomatous subjects. Furthermore, the severity of glaucoma in screened cases is objectively graded by analyzing the thickness of these regions. Results: The proposed framework is rigorously tested on publicly available Armed Forces Institute of Ophthalmology (AFIO) dataset, where it achieved the F1 score of 0.9577 for diagnosing glaucoma, a mean dice coefficient score of 0.8697 for extracting the RGC regions and an accuracy of 0.9117 for grading glaucomatous progression. Furthermore, the performance of the proposed framework is clinically verified with the markings of four expert ophthalmologists, achieving a statistically significant Pearson correlation coefficient of 0.9236. Conclusion: An automated assessment of RGC degeneration yields better glaucomatous screening and grading as compared to the state-of-the-art solutions. Significance: An RGC-aware system not only screens glaucoma but can also grade its severity and here we present an end-to-end solution that is thoroughly evaluated on a standardized dataset and is clinically validated for analyzing glaucomatous pathologies.