This paper proposes a fast method for orbit uncertainty propagation and estimation. The proposed method is based on an orbit deviation propagation approach, which consists of an analytical two-body deviation propagation solution and a deep neural network (DNN) to compensate for the errors between the two-body and the true solutions. First, five types of sample forms for training the DNN are investigated, and the optimal one is selected through learning feature and training performance analyses. Then, an uncertainty propagation solution for propagating the mean and covariance is formulated by combining the DNN-based deviation propagation approach with an unscented transformation process. Finally, a more efficient version of the unscented Kalman filter (UKF) for orbit estimation is developed based on the formulated uncertainty propagation solution. The advantage of the proposed DNN-based method is that it avoids the integration of the state transition matrix or dozens of sigma points. The performance of the proposed method is investigated on a low-Earth-orbit example. Numerical results show that the proposed DNN-based estimation method can be one order of magnitude faster than the UKF and is comparable to the UKF in estimation accuracy. In addition, it estimates more accurately than the extended Kalman filter (EKF) and is approximately 10% faster than the EKF.