We propose a new 3D inversion scheme to invert the near- and transition-zone data of CSAMT with topography accurately. In this new method, the earth was discretized into unstructured tetrahedra to fit the ragged topography and the vector finite-element method was adopted to obtain precise responses and good sensitivity. To simulate the attitude and shape of the transmitter, we divided a long-grounded transmitter into dipoles and integrated these dipoles to obtain good responses in the near- and transition-field zones. Next, we designed an L2 norm-based objective functional and applied a standard quasi-Newton method as the optimization method to solve the inverse problem and guarantee steady convergence. We tested our 3D inversion method first on synthetic data and then on a field dataset acquired from select sites near Changbai Mountain, China. In both tests, the new inversion algorithm achieved excellent fitting between the predicted and observed data, even in near- and transition-field zones, and the inversion results agreed well with the true model. These findings reveal that the proposed algorithm is effective for 3D inversion of CSAMT data.