Early and reliable identification of Alzheimer’s disease (AD) from Magnetic Resonance Imaging (MRI) remains challenging because disease-related structural changes are often subtle, spatially distributed, and difficult to model with conventional feature extractors. This study proposes an explainable Swin Transformer and Temporal Convolutional Network (Swin-TCN) framework for MRI-based AD classification. The Swin Transformer is used to learn hierarchical spatial representations from preprocessed brain MRI volumes, while the Temporal Convolutional Network models ordered inter-slice dependencies without the computational burden of recurrent architectures. A Hybrid Attention Mechanism (HAM) further recalibrates channel-level and slice-level information, allowing the model to focus on diagnostically relevant patterns. Differential Evolution (DE) is adopted to optimize key hyperparameters, including the learning rate, batch size, kernel size, dilation rate, and regularization settings. The framework was evaluated on the publicly available Kaggle MRI and Alzheimer’s dataset using subject-wise five-fold cross-validation to reduce the risk of slice-level data leakage. Under the same preprocessing pipeline and data splits, the proposed model outperformed CNN-, recurrent-, and transformer-based baselines, achieving 99.50% accuracy and an area under the curve (AUC) of 0.92. Grad-CAM++ and Integrated Gradients were used to visualize brain regions contributing to the model predictions. These results suggest that the proposed framework provides a high-performing and interpretable approach for MRI-based AD classification, although further validation on external clinical datasets is required before clinical deployment.