In this research article, we present convenient analytical-approximate solutions for fluid flow models known as multi-dimensional Navier-Stokes equations containing time-fractional order by using a relatively new analytical method called modified generalized Mittag-Leffler function method. The Caputo fractional derivative is used to describe fractional mathematical formalism. The approximate solutions for five problems are implemented to demonstrate the validity and accuracy of the proposed method. It is also demonstrated that the solutions obtained from our method when $α = 1$ coincide with the exact solutions, this is displayed by using some 2D and 3D plots for each problem. Moreover, the comparison between our outcomes with given exact solutions and results obtained by other methods in the literature besides absolute error is provided in some tables. Additionally, we offer some plots when $α$ has different values to present the effect of fractional order on the solution of each suggested problem. The numerical simulation presented in this work indicates that the proposed method is efficient, reliable, accurate and easy which has less computational ability to give analytical-approximate solution form. So, this method can be extended to implement on different related problems arising in various areas of innovation and research.