The COVID-19 pandemic has had a significant negative impact on the world in various ways. In an effort to mitigate the negative effects of the pandemic, this study proposes a deep learning approach for the automatic detection of COVID-19 from chest computed tomography (CT) images. This would enable healthcare professionals to more efficiently identify the presence of the virus and provide appropriate care and support to infected individuals. The proposed deep learning approach is based on binary classification and utilizes members of the pre-trained EfficientNet model family. These models were trained on a dataset of real patient images, called the EFSCH-19 dataset, to classify chest CT images as positive or negative for COVID-19. The results of the predictions made on the test images showed that all models achieved accuracy values of over 98%. Among these models, the EfficientNet-B2 model performed the best, with an accuracy of 99.75%, sensitivity of 99.50%, specificity of 100%, and an F1 score of 99.75%. In addition to the high accuracy achieved in the classification of chest CT images using the proposed pre-trained deep learning models, the gradient-weighted class activation mapping (Grad-CAM) method was also applied to further understand and interpret the model's predictions.