The interaction of femtosecond laser with plasmas can produce ultrashort and brilliant extreme-ultraviolet (EUV) radiations, which can be applied in high-resolution imaging and time-resolved spectroscopy. To further improve the radiance, the high-efficiency generation of EUV radiations from carbon nanotube foam (CNF) targets irradiated by a relativistic femtosecond laser is demonstrated. The experimental result indicates that when the laser energy is 1.2 J and the CNF density is 4 mg/cm ³ , the single-shot EUV spectrum intensity is around 0.1 mJ·nm ^-1·sr ^-1. Compared with high-density solid targets, the low-density CNF targets can effectively improve laser absorptivity, and therefore realize an enhancement of EUV conversion efficiency by two orders of magnitudes. Furthermore, the EUV radiations from CNFs are quasi-continuum in the wavelength range of 15--30 nm, which is suitable for applications including ultrafast absorption spectroscopy.