Brain tumor segmentation from Magnetic Resonance Imaging is a critical step for diagnosis and treatment planning, yet manual delineation is time-consuming and subjective. While deep learning models like Convolutional Neural Networks and Transformers have advanced automated segmentation, they face inherent limitations. CNNs struggle with long-range dependencies, while pure Transformers are often data-inefficient. To address these challenges, we propose DASA-Net, a novel hybrid Dual-Attention Swin-Transformer Architecture for precise brain tumor segmentation. Our model integrates a pre-trained Swin Transformer as the encoder to capture rich hierarchical and global contextual features. We introduce an Enhanced Spatial Attention module with residual connections to refine spatial details in the encoder's feature maps, and incorporate Squeeze-and-Excitation blocks in the decoder for adaptive channel-wise feature recalibration. Additionally, a Dynamic Hybrid Loss function is employed to balance Binary Cross-Entropy, Dice, and Focal losses, effectively addressing class imbalance and boundary ambiguity. Extensive experiments on the BRISC and Figshare brain tumor datasets demonstrate that DASA-Net achieves state-of-the-art performance, with a Dice score of 90.3% and IoU of 82.3% on BRISC, significantly outperforming existing methods.