One of the long-standing and challenging problems in microwave engineering is the realization of ultra-wideband absorption using extremely-thin structures. Magnetic material can facilitate thickness reduction for microwave absorbers but also bring inherent narrowband admittance matching conundrum originating from its frequency-dispersive permeability and high permittivity. In this paper, we propose a simple and yet effective solution based on the concept of admittance matching with non-Foster metasurface (NFMS). Building on this concept, an ultra-wideband and extremely-thin magnetic absorber is achieved, with a simple structure consisting of a conductor-backed magnetic sheet (CMBS) coated by a NFMS. The NFMS with negatively inductive susceptance can properly cancel its positively frequency-dispersive counterpart from the CMBS so that constructive interference near the absorber can be obtained over a wide frequency band. Furthermore, the NFMS will compensate the surface conductance required for maximum incident power dissipation. As an example, we demonstrate an absorber with one-frequency decade bandwidth and a thickness of only 1/255 wavelength at the lowest operation frequency. The proposed concept enables versatile admittance matching techniques using a single-layered and has the potential to be used in the development of interesting low-profile and broadband microwave devices.