Aston Villa defender Ron Vlaar stepped up but goalkeeper Sergio Romero came back to haunt the Dutch manager as Argentina went on to win 4-2.
Romero denied Vlaar and also Wesley Sneijder from 12 yards to secure Argentina a showdown with Germany in Sunday’s final.
Van Gaal's team were favourites to win the shoot-out after defeating Costa Rica on penalties in the quarter-finals.
"It should give you confidence. We took those penalties in an incredible way against Costa Rica," Van Gaal said.
"But the issue is you score the first one and I asked two players to take the first ball before ending up with Vlaar.
"I thought he was the best player on the pitch so should have a lot of confidence.
"It just goes to show it's not easy scoring in a penalty shoot-out.
"It's the most terrible scenario, to lose on penalties: at the very least we were equal with them, if not the better team.
"It's a big disappointment,” admitted van Gaal, who knows all about goalkeeper Romero.
The Netherlands coach signed him for AZ Alkmaar from Buenos Aires-based Racing Club in 2007 and the Argentinian spent two years working under van Gaal before the Holland coach departed for Bayern Munich.
"I taught Romero how to stop penalties (at Alkmaar), so that hurts," added van Gaal.
"We were the club to bring him to Europe.
"He was a big talent, and someone who has the qualities to do that."
Arjen Robben was a broken a man after his side's World Cup shoot-out defeat.
Robben, who calmly slotted in his spot-kick, told Dutch TV: "It hurts but we gave it our all this evening and it's grim going out in this way."
Team-mate Vlaar, who produced a defending masterclass in the middle to shut out the Argentine attack, was downbeat after missing the opening penalty in the shoot-out.
The Aston Villa star said: "I would never walk away from my responsibility. I wasn't nervous, I was focused. But it had to go in and it didn't.
"Unbelievable. It's tough, but that's what sport is about."