The 22-year-old did not even make it to half-time in the 3-3 draw against Paraguay and he has since been ruled out of Dortmund's clash with Braunschweig after aggravating a back problem.
Dortmund officials have criticised Germany coach Joachim Low for picking him when he was not fully fit.
Although Klopp is disappointed to be without a player who played such an eye-catching role in last weekend's 4-0 win over Augsburg, he has not shared his club's criticism of the player's decision to play in midweek.
"It's very easy to say in hindsight that he should not have played," Klopp said. "But I'd prefer to have somebody who gives it a go anyway and then realises that it's not right than somebody who, after sneezing once, says they need a four-week break in the Alps.
"Ilkay doesn't have 500 caps under his belt and he wants to get every single one he can, and now he just happens to be injured.
"We are going to lose other people over the course of the season and the reason really doesn't matter.
"Our job is to deal with it."
The blow has been softened slightly by the return of summer signing Henrikh Mkhitaryan from injury.
The Armenian is in line for his first competitive appearance for Dortmund tomorrow, although maybe not from the start.
"I don't know about that, we'll have to test it out so I can see," said Klopp at a press conference.
"He's getting better by the minute, but we'll have to wait and see how we start."
Braunschweig coach Torsten Lieberknecht is more concerned about how his team will finish the game in the Westfalenstadion.
The league newcomers lost their opening game of the season at home to Werder Bremen a week ago and are under no illusions about how hard their task is going to be in Westphalia.
"We're realistic and we'll just try to make as good an impression of ourselves as possible and call on all we possibly can," said Lieberknecht, a former roommate of Klopp's during their playing days together at Mainz.
"We just got carried away a bit after an hour against Bremen and that cost us the game, but this experience serves us well."