Ottmar Hitzfeld’s side kept the Albiceleste at bay until the 118th minute of the round-of-16 clash in Sao Paulo, only for Angel Di Maria to slot home the only goal from Lionel Messi's pass.
And the Hamburg defender insists that the nature of the performance provides no comfort for the Swiss as they bow out of the finals in Brazil.
"I don't think there's anything harder in football then losing a game like that at the end. Football can be cruel sometimes," Djourou told reporters.
"If you lose maybe 3-0 or 4-0 you think 'we had no chance', but today in the first half we had a lot of chances that we didn't convert.
"I think we can be proud of what we did because we played great against Argentina. But in the end, it wasn't enough."
The former Arsenal man, who completed his permanent move to the German side in April, also paid tribute to outgoing coach Ottmar Hitzfeld, who led the team for the final time despite the loss on Monday night of his elder brother Winfried.
"He can only be proud of what he's done with us," Djourou added.
"He took a young team and we all progressed under him. He doesn't talk that much but he knows the game so well. That's the type of character you will always give everything for.
"Today we did that, and we are sad that it's his last game because we would have loved to give him a few more. But it wasn't to be today."
Team-mate Gelson Fernandes insists that the future is bright for Switzerland despite Hitzfeld's retirement, pointing to the promise of the likes of youngsters Xherdan Shaqiri and Ricardo Rodriguez.
"We have a good side, so we try to play football and to create things. We are not a small team anymore," said Fernandes.
"Xherdan Shaqiri was superb today. He's a great player, a great talent. He's only 22 years old, [Josip] Drmic is 22, Rodriguez is 21, [Haris] Seferovic is 22. Everybody is young.
"So I say congratulations to them, and they need to keep their heads up and in two years they can come back for the European Championship to do better."