Early identification of Alzheimer's disease (AD) is critical for optimal management and treatment. In this research, we present a unique approach combining DenseNet-121 for feature extraction from MRI images and Support Vector Machine (SVM) for classification to identify distinct phases of Alzheimer's disease. The DenseNet-121 model, recognised for its dense connection and effective feature reuse, is applied to extract high-level, discriminative features from MRI scans. These features are then used to train an SVM classifier, which is fine-tuned to obtain optimal efficiency. The suggested method has been evaluated on the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset, which includes MRI images labelled for various stages of Alzheimer's disease. Our methodology outperforms previous methods, with an accuracy of 92.5%, precision of 91.2%, recall of 94.1%, and F1-score of 92.6%. These findings suggest that pairing DenseNet-121 with SVM gives a reliable and accurate technique for detecting Alzheimer's disease. This study demonstrates the feasibility of combining deep learning for feature extraction with standard machine learning classifiers to improve diagnosis accuracy in medical imaging. The findings indicate that the proposed technology could be further refined and utilised in clinical settings for early Alzheimer's disease detection, allowing for timely intervention and care.