Accurate and robust Direction of Arrival (DOA) estimation with small microphone arrays is gaining an increasing demand in service robotics and smart home applications. Classic non-learning DOA estimation methods show unsatisfactory performance under low SNR or high reverberation conditions. Meanwhile, some research outcomes illustrate that learning methods with Neural Networks (NN) ask for careful array element quantity or layout regulation which is impractical for many applications. In order to obtain robust DOA estimation with small arrays, taking the learning ability of Deep Neural Networks (DNN), we propose to form the training pairs by using Acoustic Vector Sensor - Direction of Arrival (AVS-DOA) cues and its counterpart DOA which can be simulated under different SNR and reverberation conditions. Then DNN-based DOA model is trained accordingly and the performance of the model has been fully investigated with different activation functions, network structures and dropout rates. With the cross-validation process, the model performing best experimentally is selected as the optimal DOA model. Experimental results validate the effectiveness of our DNN based DOA model which outperforms the non-learning method, especially under poor acoustic conditions.