The battery(ESb)-supercapacitor(ESsc) hybrid energy storage system (HESS) is the most promising solution for DC microgrids (MGs) to realize the power balance, where system instability caused by the high penetration of constant power loads (CPLs) is also a critical concern. To achieve the decentralized automatic power sharing and DC bus voltage regulation of the HESS in DC MGs towards large-signal stability, a novel autonomous finite-time backstepping control (FTBC) algorithm is proposed in this article. First, taking the characteristics of the DC bus configuration caused by CPLs into consideration, the v02-P0 droop algorithm has been proposed for the HESS to realize the dynamic power sharing among energy storages (ESs) and to maintain the DC bus voltage. With the help of the proposed scheme, the ESb supplies the low-frequency part of the load fluctuations while the ESsc compensates for the high-frequency ones. Second, with the help of extended finite-time observers (EFTOs), various disturbances, such as power coupling between ES-interfaced converters and parameter uncertainties are automatically eliminated by the EFTOs in a finite time, so that the proposed controller can achieve autonomous and decentralized control with no output current sensors and communication links. Then, the FTBC can precisely offset the estimations and stabilize the DC bus voltage in the large-signal sense. Simulation and experimental results verify the effectiveness and superiority of the proposed controller.