We have studied two different kinds of terahertz first-order distributed feedback quantum cascade lasers (THz-DFB-QCL), both of which are based on a metal-metal waveguide. To realize stable single-mode emission, we have demonstrated for the first time THz-DFB-QCLs in which THz wave is extracted via a diffraction grating. Our simulations illustrate that an optimized diffraction grating will give rise to a diffraction efficiency of about 70%, and a reflectivity less than 1%. It will also result in a laser beam pattern with a low divergence angle of 10°×50°. Such a low reflectivity will keep the mode oscillation inside the DFB grating undisturbed, which is crucial to realize stable single-mode emission. Experimentally, by utilizing a diffraction grating to extract THz wave, we have realized THz-DFB-QCLs with stable single-mode emission near 2.58 THz and the side-mode-suppression-ratio is about 23 dB. The measured laser beam pattern is in good agreement with the calculations. Most importantly, due to the good diffraction efficiency and beam directionality, the diffraction grating based THz-DFB-QCL significantly improves the output power compared with the edge-emitting counterpart.