In this paper, the Bernstein collocation method (BCM) is used for the first time to solve the nonlinear magnetohydrodynamics (MHD) Jeffery–Hamel arterial blood flow issue. The flow model described by nonlinear partial differential equations is first transformed to a third-order one-dimensional equation. By using the Bernstein collocation method, the problem is transformed into a nonlinear system of algebraic equations. The residual correction procedure is used to estimate the error; it is simple to use and can be used even when the exact solution is unknown. In addition, the corrected Bernstein solution can be found. As a consequence, the solution is estimated using a numerical approach based on Bernstein polynomials, and the findings are verified by the 4th-order Runge–Kutta results. Comparison with the homotopy perturbation method shows that the present method gives much higher accuracy. The accuracy and efficiency of the proposed method were supported by the analysis of variance (ANOVA) and 95% of confidence on interval error. Finally, the results revealed that the MHD Jeffery–Hamel flow is directly proportional to the product of the angle between the plates α and Reynolds number Re .