Objective: This study aimed to design and systematically evaluate an architecture, proposed as the Explainable Mandibular Third Molar Convolutional Neural Network (E-mTMCNN), for detecting the presence of mandibular third molars (m-M3) in panoramic radiography (PR). The proposed architecture seeks to enhance the accuracy of early detection and improve clinical decision-making and treatment planning in dentistry. Methods: A new dataset, named the Mandibular Third Molar (m-TM) dataset, was developed through expert labeling of raw PR images from the UESB dataset. This dataset was subsequently made publicly accessible to support further research. Several advanced image preprocessing techniques, including Gaussian filtering, gamma correction, and data augmentation, were applied to improve image quality. Various Deep learning (DL) based Convolutional Neural Network (CNN) architectures were trained and validated using Transfer Learning (TL) methodologies. Among these, the E-mTMCNN, leveraging the GoogLeNet architecture, achieved the highest performance metrics. To ensure transparency in the model's decision-making process, Local Interpretable Model-Agnostic Explanations (LIME) were integrated as an eXplainable Artificial Intelligence (XAI) approach. Clinical reliability and applicability were assessed through an expert survey conducted among specialized dentists using a decision support system based on the E-mTMCNN. Results: The E-mTMCNN architecture demonstrated a classification accuracy of 87.02%, with a sensitivity of 75%, specificity of 94.73%, precision of 77.68%, an F1 score of 75.51%, and an area under the curve (AUC) of 87.01%. The integration of LIME provided visual explanations of the model's decision-making rationale, reinforcing the robustness of the proposed architecture. Results from the expert survey indicated high clinical acceptance and confidence in the reliability of the system. Conclusion: The findings demonstrate that the E-mTMCNN architecture effectively detects the presence of m-M3 in PRs, outperforming current state-of-the-art methodologies. The proposed architecture shows considerable potential for integration into computer-aided diagnostic systems, advancing early detection capabilities and enhancing the precision of treatment planning in dental practice.