We present the application of a deep machine learning technique to classify radio images of extended sources on a morphological basis using convolutional neural networks (CNN). In this study, we have taken the case of the Fanaroff-Riley (FR) class of radio galaxies as well as radio galaxies with bent-tailed morphology. We have used archival data from the Very Large Array (VLA)-Faint Images of the Radio Sky at Twenty Centimeters survey and existing visually classified samples available in the literature to train a neural network for morphological classification of these categories of radio sources. Our training sample size for each of these categories is ~200 sources, which has been augmented by rotated versions of the same. Our study shows that CNNs can classify images of the FRI and FRII and bent-tailed radio galaxies with high accuracy (maximum precision at 95%) using well-defined samples and a "fusion classifier," which combines the results of binary classifications, while allowing for a mechanism to find sources with unusual morphologies. The individual precision is highest for bent-tailed radio galaxies at 95% and is 91% and 75% for the FRI and FRII classes, respectively, whereas the recall is highest for FRI and FRIIs at 91% each, while the bent-tailed class has a recall of 79%. These results show that our results are comparable to that of manual classification, while being much faster. Finally, we discuss the computational and data-related challenges associated with the morphological classification of radio galaxies with CNNs.