The Liverpool keeper denied Lukas Marecek, Jan Moravek and Radim Reznik during the shoot-out to send Hungary through to the last eight of a FIFA competition for the first time since the 1985 U-17 World Cup, and leave Czech Republic, the runners-up in 2007, heading for home.
This was an even game in which both sides took turns to dominate play. Adam Presinger struck the woodwork with an intelligent free-kick in the early stages and with Hungary asking the questions, they took advantage of their dominance in the 15th minute when Mate Kiss hit a stunning right-foot shot which flew into the top corner of the net.
However, Czech Republic were level 11 minutes later as they took their first real chance of the game. Tomas Fabian put in a cross from the left and Jan Vosahlik was in the right place at the right time to glance a close range header past Gulacsi.
It was all Czech Republic for the remainder of the first half, as they dominated their European neighbours in terms of territory and possession. Marecek was disappointed with himself as he blazed a shot well over the crossbar from the edge of the box.
Hungary thought that they had taken the lead for the second time in the match, but scorer Vosahlik headed the ball off the line to deny Vladimir Koman what would have been a certain goal. Fabian forced Gulacsi into a fine save with a low shot, while Vosahlik also saw a header go off-target.
Krisztian Nemeth created a good opportunity for himself, but dragged his shot wide, while Marecek was wasteful once again when he fired over after substitute Michael Rabusic had set him up as extra time beckoned.
It took just two minutes for the Czechs to take the lead in the first period of extra time. Martin Zeman put a good cross into the box, which Vosahlik headed on for Rabusic to power the ball into the net. Yet, seven minutes later, Hungary drew level with a goal as every bit as special as their first.
A foul on Nemeth gave Hungary a free-kick in a dangerous position and Koman was determined to take the resulting set piece. The captain stood over the ball and hit a precise right footed shot, which curled over the wall and dipped just beyond the despairing dive of Vaclik.
Reznik should have made it 3-2 for Czech Republic in the 107th minute. Jan Lecjacks crossed from the left, which evaded the Hungarian defence and dropped nicely for the substitute ten yards from goal. He had time to place his shot, but rushed his effort, which trickled wide of the far post. That was the game's last major chance and soon afterwards, it was time for penalties.
Hungary's goalscorers Presinger and Koman both scored from the spot, as did Andras Gosztonyi, while Rabusic, Ondrei Celustka and Vosahlik were on target for the Czechs, but thanks to two misses apiece the shoot-out moved into sudden death. Gulasci produced a superb one-handed save to keep Reznik's effort out and when Adam Balajti sent Vaclik the wrong way, these two formidable European sides had finally been separated.