New developments in manufacturing processes impose the need for experimental studies concerning the determination of beneficial process-related parameter settings and optimization of objectives related to quality and efficiency. This work aims to improve cutting geometry, surface texture, and arithmetic surface roughness average in the case of post-processing of filament material extrusion 3D-printed acrylonitrile styrene acrylate (ASA) thin plates by a low-power CO2 laser cutting apparatus. This material was selected owing to its unique properties for thin-walled customized constructions. Three parameters, namely focal distance, plate thickness, and cutting speed, were examined with reference to the Box-Behnken design of experiments (BBD) and regression modeling. Four responses were considered: mean kerf width, Wm (mm); down width, Wd (mm); upper width, Wu (mm); and average surface roughness Ra (μm) of cut surfaces. Different regression models were tested for their efficiency in terms of predicting the objectives with an emphasis on full quadratic regression. The results showed that a focal distance of 6.5 mm and 16 mm/s speed optimizes all quality metrics for the three plate thicknesses. The regression models achieved adequate correlation among independent process-related parameters and optimization objectives, proving that they can be used to improve the laser cutting process and support practical applications.