We present the large-scale sidereal anisotropy of Galactic cosmic-ray intensity in the multi-TeV region observed with the Tibet-III air shower array during the period from 1999 through 2003. The sidereal daily variation of cosmic rays observed in this experiment shows an excess of relative intensity around 4-7 hr local sidereal time as well as a deficit around 12 hr local sidereal time. While the amplitude of the excess is not significant when averaged over all declinations, the excess in individual declination bands becomes larger and clearer as the viewing direction moves toward the south. The maximum phase of the excess intensity changes from &SIM; 7 hr at the Northern Hemisphere to &SIM; 4 hr at the equatorial region. We also show that both the amplitude and the phase of the first harmonic vector of the daily variation are remarkably independent of primary energy in the multi-TeV region. This is the first result determining the energy and declination dependences of the full 24 hr profiles of the sidereal daily variation in the multi-TeV region with a single air shower experiment.