The number of vehicles and turning movements at roundabouts provide important information for planning, design and operational analysis of roundabouts. The visual data collected through video cameras make it possible to determine such information via computer-based methods. In this work, a method for detecting, counting, and tracking vehicles in roundabout videos is proposed. There are two main contributions of the method, (i) only the moving vehicles are considered for tracking (moving vehicle detection) and (ii) the vehicle tracks output by the object tracking algorithms are processed to reduce the false track rate (trajectory union). The vehicle detection is performed using YOLOv4, and vehicle tracking throughout the video is accomplished by either Kalman filter or DeepSORT algorithm. The output of the proposed method is compared with both manual counting results and benchmark tracking results where the entry/exit matrix is generated using only YOLOv4 and an object tracker. In a 20-min video with 297 vehicles, absolute error reached by the proposed method is 14 vehicles which corresponds to normalized absolute error percentage of 1.571%. In the same video, the same error metrics obtained by the benchmark method are 33 vehicles and 3.704%. This error tends to increase together with the rate of vehicles in the video or number of legs in the roundabout. However, the rate of increment in the error is much lower than the rate of increment in the number of vehicles in the video. In addition, lower error rates are obtained when DeepSORT is used as the vehicle tracker.