Laser scribing is applied after final coating and heat-flattening of grain-oriented (GO) electrical steel to increase the efficiency of power and distribution transformers by reducing core losses. The process is based on the refinement of the magnetic domain structure. In the case of high-power continuous-wave (cw) laser beam sources, laser lines are periodically scribed perpendicular to the rolling direction with spacings of a few millimeters between individual lines. The heating and cooling cycles along the laser lines by the dissipation of the incident laser energy induce local residual stresses, which change the magnetic domain structure. Beneath reduction of core losses, it is highly desired to avoid damage of the insulating coating by the laser treatment which can be identified by the visibility of the lines. In this study, cause-and-effect relationships between the process parameters laser power, scanning speed, and line distance and corresponding changes in magnetic properties were investigated by use of a factorial design-of-experiments (DoE) approach. The experimental trials were performed on high-permeability GO electrical steel sheets with a material thickness of 0.23 mm using a multi-mode (MM) fiber laser. In addition, a thermal model of the laser scribing process was developed in order to be able to correlate the detected changes of magnetic properties as well as the visibility of the laser lines to characteristic process temperatures. The analysis shows that the laser lines become visible for the used trail conditions if the temperature maximum reaches the melting point of the base material. On the other hand, distinctive correlations between magnetic properties and processing temperatures are not obvious. However, the evaluation indicates a non-linear dependence of the core loss and related properties on the tested process parameters.