In this work, we present a learning method for both lateral and longitudinal motion control of an ego-vehicle for the task of vehicle pursuit. The car being controlled does not have a pre-defined route, rather it reactively adapts to follow a target vehicle while maintaining a safety distance. To train our model, we do not rely on steering labels recorded from an expert driver, but effectively leverage a classical controller as an offline label generation tool. In addition, we account for the errors in the predicted control values, which can lead to a loss of tracking and catastrophic crashes of the controlled vehicle. To this end, we propose an effective data augmentation approach, which allows to train a network that is capable of handling different views of the target vehicle. During the pursuit, the target vehicle is firstly localized using a Convolutional Neural Network. The network takes a single RGB image along with cars' velocities and estimates target vehicle's pose with respect to the ego-vehicle. This information is then fed to a Multi-Layer Perceptron, which regresses the control commands for the ego-vehicle, namely throttle and steering angle. We extensively validate our approach using the CARLA simulator on a wide range of terrains. Our method demonstrates realtime performance, robustness to different scenarios including unseen trajectories and high route completion. Project page containing code and multimedia can be publicly accessed here: https://changyaozhou.github.io/Autonomous-Vehicle-Pursuit/.