In the present investigation, a continuous wave fiber laser with maximum power of 400 W was used to cut a glass fiber reinforced plastic sheet of 4.56 mm thickness using Nitrogen as assisting gas. The influence processing parameters such as laser irradiance, gas pressure, and cutting speed on the cut surface quality were investigated by using response surface methodology. The different responses of laser cut surface such as upper kerf width, taper percentage along the cut depth, and heat-affected zone on the top surface were measured to analyze the influence of input process parameters on the responses. A statistical analysis on the obtained results was conducted and found that the optimum values of different input process parameters were laser irradiance: 8.28 × 105 watt/cm2, cutting speed: 600 mm/min and assisting gas pressure: 7.84 bar. The corresponding values of responses were upper kerf width: 177.4 µm, taper 0.73%, and heat-affected zone on top surface: 109.23 µm. The confirmation experiments were conducted with the obtained optimum parameter setting and observed that the predicted values and experimental values for upper kerf width, taper percentage and top surface heat-affected zone were within the error limits of 2.52%, 1.84%, and 0.45%, respectively. Furthermore, damages like loose fibers, interlayer fractures, evaporation of matrix material and fiber breakages were observed.