Pittsburgh had taken a 3-2 lead with two goals in the opening 5:50 of the second period.
The second period goal gave the Devils a 2-2 tie.
It was the Soviets, however, who scored three first period goals to take an early 3-0 lead.
Two second period power-play goals later, it was all even.
Mark Recchi tied the game with the only goal of the second period.
Nieuwendyk scored the only goal of the second period, on a power play.
Pittsburgh led by 1-0 at both intermissions and pulled away with two goals in the first seven minutes of the third period.
Fleury's goal, at 3:25 of the first period, was his fifth in the last three games.
The Czechs replied with a second period goal and another early in the third to tie the game.
Yashin scored the winning goal at 10:17 of the second period.