For systems of polynomial equations, we study the problem of computing the Newton polytope of their eliminants. As was shown by Esterov and Khovanskii, such Newton polytopes are mixed fiber polytopes of the Newton polytopes of the input equations. We use their results in combination with mixed subdivisions to design an algorithm computing these special polytopes. We demonstrate the increase in practical performance of our algorithm compared to existing methods using tropical geometry and discuss the differences that lead to this increase in performance. We also demonstrate an application of our work to differential elimination.