Croatia, the 2018 runners-up, stunned Brazil 4-2 on penalties after Marquinhos and Rodrygo failed to convert.
Brazil's joint all-time top scorer Neymar - who was fifth in line in the shootout - did not even get the opportunity to take his penalty after his teammates missed.
"We were freaking out, it was real drama. We tried to discuss the difference between the drama from the Brazil game compared to the Argentina v Netherlands game and we came up with some ideas," Klinsmann said at a Technical Study Group briefing.
"After conceding the equaliser in the 117th minute, Brazil had no time to switch gears. From the disappointment of conceding a last-minute goal to the penalty shootout, there was no time to settle and approach the shootout positively.
"For Brazil, they should have maybe put their best shooter first, set the tone with the best penalty taker you have and then go down the rankings."
That was not the case with their South American rivals Argentina, however, who recovered from the shock of conceding a last-gasp equaliser in normal time to the Netherlands before prevailing 4-3 in the shootout.
Lionel Messi was the first to step up for Argentina and converted his penalty with aplomb, immediately putting pressure on the Dutch as Emiliano Martinez saved the first two Dutch penalties from Virgil van Dijk and Steven Berghuis.
"With Argentina, they had extra time to play, even with the shock of the (Wout) Weghorst equaliser in the last minute," Klinsmann added.
"They had 30 minutes more to play. At the end of the game, they were far more clearer (in the head) going into the penalty shootout."
Klinsmann also praised England's quarter-final performance against reigning champions France, saying the positive approach from Gareth Southgate's side will take them far with the young talent they have coming through the ranks.
"Overall it was very positive from England. France v England should have been a semi-final or final. Unfortunately one of them had to go home," Klinsmann said.
"It's still a team in growth, this is a team that can get better over the years. They have a lot of talent coming through, they're getting better and more experienced after three positive tournaments. There's more to come from this England side."
