In this article, a novel variable-bandwidth extended state observer (VBESO) is proposed to estimate external disturbances for a class of $n$th-order nonlinear systems with measurement noise. Based on the feature of “high error, high bandwidth; low error, low bandwidth,” the bandwidth of the proposed observer is adjusted automatically with the output estimation error, achieving a tradeoff between the estimation accuracy and the robustness to measurement noise. Moreover, a designed filter is introduced into the observer such that an arbitrarily small initial bandwidth can be set to alleviate the “peaking phenomenon,” and the convergence speed is improved by reducing the time constant of the filter. In addition, extensions to other kinds of observers are also established. Finally, the VBESO-based state-feedback control is presented and applied to the drag-free spacecraft hardware-in-loop experiment platform. Simulation and experiment results show that the proposed VBESO can improve the tolerance to the measurement noise and achieve a tradeoff between transient-state and steady-state performance.