This article employs the Laplace residual power series approach to study nonlinear systems of time-fractional partial differential equations with time-fractional Caputo derivative. The proposed technique is based on a new fractional expansion of the Maclurian series, which provides a rapid convergence series solution where the coefficients of the proposed fractional expansion are computed with the limit concept. The nonlinear systems studied in this work are the Broer-Kaup system, the Burgers’ system of two variables, and the Burgers’ system of three variables, which are used in modeling various nonlinear physical applications such as shock waves, processes of the wave, transportation of vorticity, dispersion in porous media, and hydrodynamic turbulence. The results obtained are reliable, efficient, and accurate with minimal computations. The proposed technique is analyzed by applying it to three attractive problems where the approximate analytical solutions are formulated in rapid convergent fractional Maclurian formulas. The results are studied numerically and graphically to show the performance and validity of the technique, as well as the fractional order impact on the behavior of the solutions. Moreover, numerical comparisons are made with other well-known methods, proving that the results obtained in the proposed technique are much better and the most accurate. Finally, the obtained outcomes and simulation data show that the present method provides a sound methodology and suitable tool for solving such nonlinear systems of time-fractional partial differential equations.