In this paper, lap joint of 2 mm thickness AA1050 sheets and C10100 sheets using different process parameters were investigated. Experimental plan is designed according to response surface methodology (RSM) where: tool rotation speed, plunge depth and dwell time vary between 500 and 2500 rpm, 2.1 and 3.7 mm and 0 and 8 s respectively. The main affecting and interaction process parameters are evaluated using analysis of variance (ANOVA) methodolgy. Mathematical models describing the relationship between welding parameters and shear failure load (SFL) are developed and the optimal FSSW parameters are specified. Design expert software is used for optimization of friction stir spot welding process parameters using response graphs and contour plots. With 95% confidence level, shear failure load of the dissimilar Al 1050-Cu 10100 joints are predicted using the developed empirical relations. It is found that the optimal results of the SFL is close to the optimal solution obtained from the mathematical model with less than 4% discrepancy.