We report on the terahertz (THz) quantum cascade lasers in continuous-wave (CW) operation with an emitting frequency above 5 THz. Excellent performance with a smaller leakage current and higher population inversion efficiency is obtained by one-well bridged bound-to-continuum hybrid quantum design at 5 THz. By designing and fabricating a graded metallic sampled distributed feedback grating in the waveguide, the first single-mode THz quantum cascade laser at 5.13 THz in CW operation mode is achieved. The maximum single-mode optical power of ∼48mW is achieved at 15 K with a side-mode suppression ratio above 24 dB. This will draw great interest in the spectroscopy applications above the 5 THz range for THz quantum cascade lasers.