Accurate and efficient classification of Alzheimer's disease (AD) severity from brain magnetic resonance imaging (MRI) remains a critical challenge, particularly when limited data and model interpretability are of concern. In this work, we propose TDA-Alz, a novel framework for four-stage Alzheimer's disease severity classification (non-demented, moderate dementia, mild, and very mild) using topological data analysis (TDA) and ensemble learning. Instead of relying on deep convolutional architectures or extensive data augmentation, our approach extracts topological descriptors that capture intrinsic structural patterns of brain MRI, followed by feature selection to retain the most discriminative topological features. These features are then classified using an ensemble learning strategy to achieve robust multiclass discrimination. Experiments conducted on the OASIS-1 MRI dataset demonstrate that the proposed method achieves an accuracy of 98.19% and an AUC of 99.75%, outperforming or matching state-of-the-art deep learning--based methods reported on OASIS and OASIS-derived datasets. Notably, the proposed framework does not require data augmentation, pretrained networks, or large-scale computational resources, making it computationally efficient and fast compared to deep neural network approaches. Furthermore, the use of topological descriptors provides greater interpretability, as the extracted features are directly linked to the underlying structural characteristics of brain MRI rather than opaque latent representations. These results indicate that TDA-Alz offers a powerful, lightweight, and interpretable alternative to deep learning models for MRI-based Alzheimer's disease severity classification, with strong potential for real-world clinical decision-support systems.