In the present work, an Elzaki transformation is combined with a decomposition technique for the solutions of fractional dynamical systems. The targeted problems are related to the systems of fractional partial differential equations. Fractional differential equations are useful for more accurate modeling of various phenomena. The Elzaki transform decomposition method is implemented in a very simple and straightforward manner to solve the suggested problems. The proposed technique requires fewer calculations and needs no discretization or parametrization. The derivative of fractional order is represented in a Caputo form. To show the conclusion, which is drawn from the results, some numerical examples are considered for their approximate analytical solution. The series solutions to the targeted problems are obtained having components with a greater rate of convergence toward the exact solutions. The new results are represented by using tables and graphs, which show the sufficient accuracy of the present method as compared to other existing techniques. It is shown through graphs and tables that the actual and approximate results are very close to each other, which shows the applicability of the presented method. The fractional-order solutions are in best agreement with the dynamics of the given problems and provide infinite choices for an optimal solution to the suggested mathematical model. The novelty of the present work is that it applies an efficient procedure with less computational cost and attains a higher degree of accuracy. Furthermore, the proposed technique can be used to solve other nonlinear fractional problems in the future, which will be a scientific contribution to research society.