The epicardial fat plays a key role in the development of many cardiovascular diseases. It is necessary and useful to precisely segment this fat from CT scans in clinical studies. However, it is not feasible to manually segment this fat in clinical practice, as the workload and cost for technicians or physicians is quite high. In this work, we propose a novel method for automatic segmentation and quantification of epicardial fat from CT scans accurately. In detail, dual U-Nets with the morphological processing layer is used for this goal. The first network is based on the U-Net framework to detect the pericardium, before segmenting its inside region. A morphological layer is concatenated as the following layer of the first network, to refine and obtain the ideal inside region of the pericardium. While the second network is also applied using U-Net as its backbone to find and segment the epicardial fat of the processed inside region from the pericardium using the first network. Our proposed method obtains the highest mean Dice similarity (91.19%), correlation coefficient (0.9304) compared to other state-of-art methods on a cardiac CT dataset with 20 patients. The results indicate our proposed method is effective for quantifying epicardial fat automatically.