Automatic segmentation of retinal images is an important task in computer-assisted medical image analysis for the diagnosis of diseases such as hypertension, diabetic and hypertensive retinopathy, and arteriosclerosis. Among the diseases, diabetic retinopathy, which is the leading cause of vision detachment, can be diagnosed early through the detection of retinal vessels. The manual detection of these retinal vessels is a time-consuming process that can be automated with the help of artificial intelligence with deep learning. The detection of vessels is difficult due to intensity variation and noise from non-ideal imaging. Although there are deep learning approaches for vessel segmentation, these methods require many trainable parameters, which increase the network complexity. To address these issues, this paper presents a dual-residual-stream-based vessel segmentation network (Vess-Net), which is not as deep as conventional semantic segmentation networks, but provides good segmentation with few trainable parameters and layers. The method takes advantage of artificial intelligence for semantic segmentation to aid the diagnosis of retinopathy. To evaluate the proposed Vess-Net method, experiments were conducted with three publicly available datasets for vessel segmentation: digital retinal images for vessel extraction (DRIVE), the Child Heart Health Study in England (CHASE-DB1), and structured analysis of retina (STARE). Experimental results show that Vess-Net achieved superior performance for all datasets with sensitivity (Se), specificity (Sp), area under the curve (AUC), and accuracy (Acc) of 80.22%, 98.1%, 98.2%, and 96.55% for DRVIE; 82.06%, 98.41%, 98.0%, and 97.26% for CHASE-DB1; and 85.26%, 97.91%, 98.83%, and 96.97% for STARE dataset.