Due to the advancement of non-invasive medical imaging modalities like Magnetic Resonance Angiography (MRA), an increasing number of Intracranial Aneurysm (IA) cases are being reported in recent years. The IAs are typically treated by so-called endovascular coiling, where blood flow in the IA is prevented by embolization with a platinum coil. Accurate quantification of the IA Remnant (IAR), i.e. the volume with blood flow present post treatment is the utmost important factor in choosing the right treatment planning. This is typically done by manually segmenting the aneurysm remnant from the MRA volume. Since manual segmentation of volumetric images is a labour-intensive and error-prone process, development of an automatic volumetric segmentation method is required. Segmentation of small structures such as IA, that may largely vary in size, shape, and location is considered extremely difficult. Similar intensity distribution of IAs and surrounding blood vessels makes it more challenging and susceptible to false positive. In this paper we propose a novel 3D CNN architecture called Dual-Attention Atrous Net (DAtt-ANet), which can efficiently segment IAR volumes from MRA images by reconciling features at different scales using the proposed Parallel Atrous Unit (PAU) along with the use of self-attention mechanism for extracting fine-grained features and intra-class correlation. The proposed DAtt-ANet model is trained and evaluated on a clinical MRA image dataset (prospective research project, approved by the local ethical committee) of IAR consisting of 46 subjects, annotated by an expert radiologist from our group. We compared the proposed DAtt-ANet with five state-of-the-art CNN models based on their segmentation performance. The proposed DAtt-ANet outperformed all other methods and was able to achieve a five-fold cross-validation DICE score of $0.73\pm0.06$.