Millimeter-wave (mmWave) communications have attracted considerable interest due to the massive available spectrum. However, to establish communication links, a beam training procedure is indispensable. How to accelerate the beam training process is one of the key challenges towards realizing mmWave communications in practice. In this study, we first propose a novel low-cost digital beamforming (DBF) module assisted hybrid (DA-hybrid) architecture, by exploiting both the capabilities of analog and digital modules. To make this topology practical, we deploy coarse radio frequency (RF) chains and low-resolution analog-to-digital converters in the low-cost DBF module to reduce cost and power consumption. Second, we design a fast beam training method (named DAH-BT) by utilizing the proposed DAhybrid architecture and leveraging the sparse nature of mmWave channels, in which an internal calibration method is adapted to obtain the parameters of the RF impairments and the orthogonal matching pursuit algorithm is utilized to estimate beams. We also prove that the developed measurement matrices satisfy the restricted isometry property. Extensive simulation results show that the DA-hybrid architecture can not only provide close to 100% beam matching accuracy, but also dramatically reduce the system power consumption and cost. In addition, the proposed DAH-BT scheme consumes the shortest time for beam training over the state-of-art methods with comparable spectral efficiency.