The Unmanned Aerial Vehicle (UAV) proliferation has raised many concerns, since their potential malicious usage renders UAVs as an effective tool for a number of illegal activities. Radar based counter-UAV applications provide a robust solution for UAV detection and classification. Most of the existing research, tackles the problem of UAV classification by extracting features from the time variations of the Fourier spectra. Yet, these solutions require that the UAV is illuminated by the radar for a longer time which can be only met by a tracking radar architecture. On the other hand, surveillance radar architectures don't have such a cumbersome requirement and are generally superior in maintaining situational awareness, due their ability for constantly searching on a 360* area for targets. Nevertheless, the available automatic UAV classification methods for this type of radar sensors are relatively inefficient. This work proposes the incorporation of the deep learning paradigm in the classification pipeline, to provide an alternative UAV classification method that can handle data from a surveillance radar. Therefore, a Deep Neural Network (DNN) model is employed to discern between UAVs and negative examples (e.g. birds, noise, etc.). Extensive experiments demonstrate the validity of the proposed method, where the overall classification accuracy can reach up to 95.0%.