Deoxidized oxygen free copper C12200, 1 mm in thickness, was welded to 1-mm thick AISI 304 stainless steel with disk laser. The butt-welded joints were produced with different welding parameters. Full factorial design of experiment (DoE) approach consisting of three factors and two levels was utilized. Laser powers used for welding were 1.3 and 1.9 kW and welding speeds of 20 and 30 mm/s. Two beam offsets were tested, namely, 100 μm toward copper side and 200 μm toward AISI 304 steel. It was found that beam offset possesses the largest influence on the welded joints’ tensile strength. Tensile strengths attained values more than 3.7 times higher in comparison to the AISI 304 steel beam offset. When lower laser power was used, the higher tensile strength was attained for copper sheet offset. Higher microhardness was observed when laser beam was offset to AISI 304 steel side. The average microhardness of the weld metal was higher than that of the weaker base material, copper sheet. Energy dispersive X-ray spectroscopy (EDS) analysis confirmed the heterogeneity in elemental composition across the welded joint interface, being lower when laser beam was offset to AISI 304 steel side. On the other hand, the copper content dropped to the average composition of weld metal at the distance of about 140 μm from copper-weld metal interface.