Retinal vascular changes are the early indicators for many progressive diseases like diabetes, hypertension, etc. However, the manual procedure in detecting these vascular changes is a time-consuming process and may cause a large variance, especially when dealing with a large dataset. Therefore, computer-aided diagnosis of the retinal vascular network plays a crucial role in analyzing the patients effectively with high precision. As a result, this paper presents a robust deep learning Multistage Dual-Path Interactive Refinement Network (DPIRef-Net) for segmenting the vascular maps of arteries and veins from the retinal surface. The main novelty of the proposed model lies in segmenting both the regional and edge salient feature maps that will reduce the degeneration problems of pooling and striding. This eventually preserves the edges of vascular branches and suppresses the false positive rate. In addition to this, a novel guided filtering technique is employed to segment the final accurate arteries and veins vascular networks from predicted regional and edge feature maps. The proposed Multistage DPIRef-Net is trained and tested on different benchmark datasets like DRIVE, HRF, AVRDB, INSPIRE AVR, VICAVR, and Dual-Mode datasets. The proposed model illustrated superior performance in segmenting the vascular maps on all datasets by achieving an average accuracy of 97%, a sensitivity of 96%, a specificity of 98%, and a dice coefficient of 98%.