Glaucoma, a progressive visual disease, damages the optic nerve, potentially resulting in irreversible vision loss. Effective treatment plans and prompt diagnosis are essential for positive patient outcomes. The principal glaucoma-related indication typically revolves around an odd ratio between the diameters of the optic cup and disc. In this regard, fundus images are analyzed for diagnosis of glaucoma. However, image quality, similar characteristics across classes, class imbalance issues, uneven shapes of optic disc and curves, and other factors make correct segmentation of optic disc and curve difficult from fundus images. To address these issues, the work proposes an enhanced Residual U-Net architecture capable of semantically segmenting the optic disc and cup from retinal fundus images. The proposed model has an Attention Module that can effectively extract the necessary feature map for segmentation from the input images. This improves the model’s capacity to pick out important features and eliminate extraneous noise. Using a publicly available dataset, the study analyses the efficacy of the proposed model using both quantitative and qualitative measures. The effectiveness of the model is examined in relation to various loss functions. To emphasize the significance of the suggested model, a comparative analysis between the proposed and the traditional model is also carried out.