With Switzerland and England level at 1-1 after extra time, Akanji missed his side's first spot kick while the Three Lions converted all five of theirs to land a place in next week's semi-final in Dortmund.
The Manchester City central defender showed his true character by volunteering to take the first penalty, Yakin said.
"You give responsibility to your players, and you can't be angry at them afterwards - players who played a fantastic tournament, especially Manu," he told a press conference.
"What can you say to him? Well, there's not many words of solace. It doesn't matter, no. I said 'Thank-you' to him for his performance, and also how he led the team, how he performed. This was really on the highest level.
"Of course, there's deep emptiness, deep sadness. But this is the kind of sadness where everybody, every individual, has to work through it."
Yakin said he was proud of the team for the way they performed at Euro 2024, denying hosts Germany a perfect start to the tournament, eliminating holders Italy and playing entertaining football.
'IT HURTS A LOT'
He said his side played well and deserved to go through after defending resolutely to keep England's big names at bay.
Breel Embolo put the Swiss ahead after 75 minutes to send travelling fans into a frenzy but the celebrations were cut short when Bukayo Saka fired a brilliant shot in off the far post to level the game.
Switzerland could have won it deep into extra time when substitute Xherdan Shaqiri rattled the woodwork with an outrageous attempt to score from a corner, before Zeki Amdouni forced a late save from Jordan Pickford.
"It hurts a lot when you make so many efforts, when you see the chances that we had. I'm very sad for the lads and also for the nation that after such a performance, we have to leave the tournament," Yakin said.
He said Akanji's missed penalty was just a detail that could be forgotten about in view of his overall performance.
"The beautiful thing is, it's only a football game. On one hand, football is giving us so much in life, you can never pay it back," he said.
"It's incredible. But these moments are part of the business. It is a missed penalty. And now, he doesn't need to be bothered anymore about it when you see how he played, the responsibility he carried on his shoulders."