Neonatal infants communicate with us through cries. The infant cry signals have distinct patterns depending on the purpose of the cries. Preprocessing, feature extraction, and feature selection need expert attention and take much effort in audio signals in recent days. In deep learning techniques, it automatically extracts and selects the most important features. For this, it requires an enormous amount of data for effective classification. This work mainly discriminates the neonatal cries into pain, hunger, and sleepiness. The neonatal cry auditory signals are transformed into a spectrogram image by utilizing the short-time Fourier transform (STFT) technique. The deep convolutional neural network (DCNN) technique takes the spectrogram images for input. The features are obtained from the convolutional neural network and are passed to the support vector machine (SVM) classifier. Machine learning technique classifies neonatal cries. This work combines the advantages of machine learning and deep learning techniques to get the best results even with a moderate number of data samples. The experimental result shows that CNN-based feature extraction and SVM classifier provides promising results. While comparing the SVM-based kernel techniques, namely radial basis function (RBF), linear and polynomial, it is found that SVM-RBF provides the highest accuracy of kernel-based infant cry classification system provides 88.89% accuracy.