The propagation properties of one-dimensional off-axis Gaussian beams through the turbulent atmosphere are studied in detail. The propagation equation of intensity is derived. It is shown that one-dimensional off-axis Gaussian beams in turbulent atmosphere undergo three stages of evolution with increasing propagation distance z, i.e., their saw-tooth beam profile is similar to the initial one in the near field, then becomes a flat-topped profile with increasing z, and at last turns into Gaussian-like profile in the far field. The turbulence accelerates the evolution of the stages which one-dimensional off-axis Gaussian beams undergo. Furthermore, the normalized intensity distributions of one-dimensional off-axis Gaussian beams with different beam numbers become close to each other due to turbulence. In addition, one-dimensional off-axis Gaussian beams with higher beam numbers are less sensitive to the effects of turbulence than those with lower beam numbers, and one-dimensional off-axis Gaussian beams are less sensitive to the effects of turbulence than Gaussian beams.