To date, exploiting the full bandwidth of state-of-the-art InGaAs photomixers for generation and detection of continuous-wave (CW) THz radiation (typ.~50 GHz to~3 THz) required complex and costly external-cavity diode lasers with motorized resonator control. Distributed feedback (DFB) lasers, by contrast, are compact and inexpensive, but the tuning range per diode is limited to~600 GHz at 1.5 μm. In this paper, we show that a combination of three DFB diodes covers the complete frequency range from 0-2750 GHz without any gaps. In combination with InGaAs-based photomixers for terahertz generation and detection, the system achieves a dynamic range of > 100 dB at 56 GHz, 64 dB at 1000 GHz, and 26 dB at 2500 GHz. A field-programmable gate array (FPGA)-based lock-in amplifier permits a flexible adjustment of the integration time from 0.5 ms to 600 ms. Employing an optimized "fast scan" mode, a spectrum of~1200 GHzthe bandwidth of each subset of two lasersand 40 MHz steps is acquired in less than one minute, still maintaining a reasonable dynamic range. To the best of our knowledge, the bandwidth of 2.75 THz presents a new record for DFB-based CW-terahertz systems.