In this paper, a new approach to enhance the radiation efficiency of microstrip patch antennas by a quarter-wavelength antireflection superstrate is proposed. It is based on dielectric right-handed metamaterial (MM) build with U-shape cells. The moments method in the spectral domain predicts a maximum radiation efficiency of 78% for disk microstrip antenna combined with quarter-wavelength dielectric antireflection superstrate. Design, optimization and characterization of the metamaterial layer are well described. A numerical analysis of the microstrip patch antenna disk with and without metamaterial antireflection layer is performed with commercial specialised software. The results of the simulation show significant performance improvement of the antenna in the presence of the metamaterial antireflection layer. Radiation efficiency, directivity and gain are respectively increased by 9%, 2dBi and 2.6dB. This approach enabled to improve the total radiated power up to 18%.