It is proved that there is, up to isotopy, a unique irreducible Heegaard splitting in an orientable, closed, connected Seifert 3-manifold with an orientable elliptical or euclidean orbifold basis. Using Hamilton's and Lawson's results, the topological uniqueness is obtained of closed orientable minimal surfaces of a given genus g [ges ] 2, embedded in a closed orientable Riemannian 3-manifold with strictly positive Ricci curvature.