The 20-year-old is Hertha's third-stringer, but got the nod in the team's final group-stage game with starter Rune Jarstein unable to play in European matches and backup Thomas Kraft out with illness.
And with Thursday's match tied 1-1 in the 87th minute, Klinsmann went down low to his left to deny Brwa Nouri from the spot.
"It was definitely crazy. I don't know how else to describe it," Klinsmann told Hertha TV after the game. "First starting XI, Europa League, the whole thing. It was just crazy.
"At the time, I was just nervous. A couple of days before the game and you start thinking and overthinking. I didn't know what to think."
On the penalty save, Klinsmann said: "It was funny, right before the game we didn't actually go over penalties until right before kick-off. It wasn't really a thing we thought would happen but obviously you have to prepare for the worst.
"I just told myself, most of these guys go left so if I go and I have a feeling, just go 100 per cent to that corner and it worked."
"Getting my first game, it's a whole different atmosphere if you're a goalkeeper. I have a lot of close relationships with the guys in the second team and then to get this win and you saw they were grateful for the save and happy for you and the team, it was great to see," he continued.
Hertha head coach Pal Dardai, whose side finished bottom of Group J, added: "Klinsmann had a very good game, he stayed firm under pressure and saved the penalty. But it wasn't just him who played well.
"Jonathan is a young man and he had a good game. But he needs to keep working hard, continue to learn and build on this showing. Today he gets an A."
Klinsmann, the son of legendary striker and former Bayern Munich, Germany and USA manager Jurgen, signed with the Bundesliga side in the summer after impressing for the United States at the U-20 World Cup in South Korea.
Since arriving in Germany, the California-born Klinsmann has been playing for Hertha's reserve side in the lower divisions of German football.