Conventional model-based sensorless surface-mounted permanent magnet synchronous motor (SPMSM) drivers suffer deteriorated estimation performance of speed and position in transient-state operation such as accelerating during wide-speed range or encountering large load torque disturbance. In this paper, an effective sensorless control solution is proposed to improve the robustness of sensorless SPMSM drivers in transient-state operation. Two extended state observers (ESOs) are utilized to estimate the αβ axis back-EMFs. An adaptive bandwidth tuning scheme is proposed to online tune the parameters of the ESOs, with which the desired estimation performance of back-EMFs can be achieved in wide-speed operation range. Additionally, the dynamic mechanical behavior of the sensorless control system during large load torque transient is investigated by using small-signal analysis, and then a corresponding speed estimation compensation scheme for the conventional quadrature phase-locked loop (QPLL) is developed, with which the capability of QPLL in tracking the rapidly-changing speed can be significantly enhanced. Experiments on a SPMSM demonstrate the superior robustness of the proposed method to that of the conventional method in transient-state operation.