Monitoring dolphins in the open sea is essential for understanding their behavior and the impact of human activities on the marine ecosystems. Passive Acoustic Monitoring (PAM) is a non-invasive technique for tracking dolphins, providing continuous data. This study presents a novel approach for classifying dolphin vocalizations from a PAM acoustic recording using a convolutional neural network (CNN). Four types of common bottlenose dolphin (Tursiops truncatus) vocalizations were identified from underwater recordings: whistles, echolocation clicks, burst pulse sounds, and feeding buzzes. To enhance classification performances, edge-detection filters were applied to spectrograms, with the aim of removing unwanted noise components. A dataset of nearly 10,000 spectrograms was used to train and test the CNN through a 10-fold cross-validation procedure. The results showed that the CNN achieved an average accuracy of 95.2% and an F1-score of 87.8%. The class-specific results showed a high accuracy for whistles (97.9%), followed by echolocation clicks (94.5%), feeding buzzes (94.0%), and burst pulse sounds (92.3%). The highest F1-score was obtained for whistles, exceeding 95%, while the other three vocalization typologies maintained an F1-score above 80%. This method provides a promising step toward improving the passive acoustic monitoring of dolphins, contributing to both species conservation and the mitigation of conflicts with fisheries.