He participated in Red Star's 4–2 win over Real Madrid in European Cup quarter-finals and stopped a Hugo Sanchez's penalty kick in Belgrade.
He won his first Yugoslav title with Red Star in the 1987-88 season and repeated it by winning a national double in 1989-90.
Although Red Star had a comfortable 2–0 advantage from the first match in Belgrade (in which Stojanović was the best player alongside Dejan Savićević), the side fell apart at Müngersdorfer Stadion and conceded two late goals before losing 3–2 on aggregate.
However, he blocked several difficult shots before Augenthaler scored an infamous own goal in injury time, which sent Red Star to Bari, to participate in the finals against Olympique de Marseille.
Again, there was no glamour from Red Star's Savićević or Prosinečki, it was Stojanović who stole the show during 120 minutes and especially in the penalty shootout, when he stopped Manuel Amoros's shot, which resulted in Red Star Belgrade triumphing 5–3 on penalties.