In this work, we have developed here a striking numerical method for solving the Burgers equation. The numerical scheme is based on collocation of the modified cubic B-splines basis functions in space variable. The obtained results have been computed without using any linearization and transformation processes. The produced diagonal system has been solved by the optimal strong stability preserving time stepping Runge-Kutta for five stage and order four scheme(SSPRK54). The present approach has been seen to be appropriate for the advection dominant cases. The effectiveness of this method has been verified by considering some test problems. The numerical solutions are in good agreement with the exact solutions and available literature. The present method has been seen to be relatively easy and economical for researchers. And also, the proposed scheme needs relatively less storage space and computational time.