Ranging has been regarded as one of the fundamental enabling technologies for a multitude of applications that require high accurate position information, such as automated navigation, vehicle platooning, asset management, etc. Among various ranging techniques, impulse-radio ultra-wideband is one of the most competitive technologies for high-precision ranging, because of its capability of achieving centimeter-level ranging accuracy, even for dense urban, indoor or cave like environments. However, two main challenges arise when fully exploiting the ranging capability of impulse-radio ultrawideband: (i) the extremely high sampling rate to acquire the received multipath signal, and (ii) the optimal thresholding strategy to differentiate the first path. To efficiently tackle those challenges, in this work, we propose a ranging approach under the compressed sensing framework. Specifically, the received ranging signal is acquired by low-rate compressed sampling through parallel random projections. Then, an algorithm named matching-pursuit search-back is proposed to detect the first arrival path, which integrates a backward iterative search and thresholding process starting from the peak path. The detection threshold is dynamically adjusted in each iteration to asymptotically minimize the averaged detection errors over false alarm and missed detection. Extensive simulations and experiments with field data are provided to demonstrate that the proposed approach can achieve high-precision ranging with far fewer samples compared with the traditional Nyquist-sampling based ones.