Impedance mismatch generally exists upon interfaces between different media. This is especially true for TE-polarized waves with large incident angles since there is no Brewster effect. As a result, high-efficiency transmission can only be guaranteed within limited incident angle range. It is desirable that transparent windows possess robust angle-stability. In this work, we propose a strategy of realizing transparent windows with extreme angle-stability using anisotropic metasurfaces. Different from traditional isotropic materials, anisotropic metasurfaces require specific three-dimensional permittivity and permeability parameters. Theoretical formulas are derived to realize a highly efficient transmission response without angular dispersion. To validate our design concept, a two-layer cascaded electromagnetic anti-reflector is designed, and it exhibits a characteristic impedance matching for nearly all incidence angles under TE-polarization illumination. As a proof-of-concept, a prototype of extremely angle-stable transparent window is fabricated and measured. Compared with the pure dielectric plate, the reflection coefficients are on average reduced by 40% at 13.5 GHz for TE-polarized waves from 0° to 80°. Therefore, we think, anisotropic cascaded electromagnetic transparent windows are capable of tailoring the electromagnetic parameter tensors as desired, and provide more adjustable degrees of freedom for manipulating electromagnetic wavefronts, which might open up a promising way for electromagnetic antireflection and find applications in radomes, IR windows and others.