Analog beamforming is a low-cost architecture for millimeter-wave (mmWave) mobile communications. However, it has two disadvantages for serving fast mobility users: (i) the mmWave beam in the wireless channel and the beam steered by analog beamforming have small angular spreads which are difficult to align with each other and (ii) the receiver can only observe the mmWave channel in one beam direction and rely on beam-probing algorithms to check other directions. In this paper, we develop a beam probing and tracking algorithm that can efficiently track fast-moving mmWave beams in three-dimensional (3D) space. This algorithm has several salient features: (1) fading channel supportive: it can simultaneously track the channel coefficient and two-dimensional (2D) beam direction in fading channel environments; (2) low probing overhead: it achieves the minimum probing requirement for joint beam and channel tracking; (3) fast tracking speed and high tracking accuracy: its tracking error converges to the minimum Cramer-Rao lower bound (CRLB) in static scenarios in theory and it outperforms several existing tracking algorithms with lower tracking error and faster tracking speed in simulations.