Haemoglobin (Hb) electrophoresis is a method of blood testing used to detect thalassaemia. However, the interpretation of the result of the electrophoresis test itself is a complex task. Expert haematologists, specifically in developing countries, are relatively few in number and are usually overburdened. To assist them with their workload, in this paper we present a novel method for the automated assessment of thalassaemia using Hb electrophoresis images. Moreover, in this study we compile a large Hb electrophoresis image dataset, consisting of 103 strips containing 524 electrophoresis images with a clear consensus on the quality of electrophoresis obtained from 824 subjects. The proposed methodology is split into two parts: (1) single-patient electrophoresis image segmentation by means of the lane extraction technique, and (2) binary classification (normal or abnormal) of the electrophoresis images using state-of-the-art deep convolutional neural networks (CNNs) and using the concept of transfer learning. Image processing techniques including filtering and morphological operations are applied for object detection and lane extraction to automatically separate the lanes and classify them using CNN models. Seven different CNN models (ResNet18, ResNet50, ResNet101, InceptionV3, DenseNet201, SqueezeNet and MobileNetV2) were investigated in this study. InceptionV3 outperformed the other CNNs in detecting thalassaemia using Hb electrophoresis images. The accuracy, precision, recall, f1-score, and specificity in the detection of thalassaemia obtained with the InceptionV3 model were 95.8%, 95.84%, 95.8%, 95.8% and 95.8%, respectively. MobileNetV2 demonstrated an accuracy, precision, recall, f1-score, and specificity of 95.72%, 95.73%, 95.72%, 95.7% and 95.72% respectively. Its performance was comparable with the best performing model, InceptionV3. Since it is a very shallow network, MobileNetV2 also provides the least latency in processing a single-patient image and it can be suitably used for mobile applications. The proposed approach, which has shown very high classification accuracy, will assist in the rapid and robust detection of thalassaemia using Hb electrophoresis images.