The USA went down 1-0 against group winners Germany in Recife on Thursday but a superior goal difference to Portugal, with whom they shared a dramatic 2-2 draw on Sunday, proved enough for Klinsmann's side to qualify for the knockout stages.
Although the former Bayern Munich and Tottenham striker conceded there were flaws in his side's performance against the country of his birth, he was delighted to have upset the odds in a tough group that also featured Ghana - the nation that has eliminated the USA from the past two World Cups.
"It's huge," Klinsmann told television reporters after Thomas Muller's 55th-minute strike settled matters. "Obviously we wanted at least a tie out of the game.
"Maybe in the beginning we had a bit too much respect but then, more and more, we got into the game.
"We should have created a few more chances, that's really something we've got to improve on.
"But overall, tremendous energy and tremendous effort from the whole side and it's huge for us, getting out of this group.
"Everybody said 'you have no chance'. We took that chance and we move on.
"Now we really want to prove a point."
The tournament's fixture schedule granted Germany an extra 24 hours' rest following their weekend exertions against Ghana.
But Klinsmann refused to view this as a factor after watching his players emerge unscathed from a sluggish start at a sodden Arena Pernambuco.
"I don't want to put too much into it," he said. "I think the energy was great, the commitment was great.
"We should have had a little bit more possession and created more chances but maybe we've kept that for the next one."
Kiinsmann shuffled his pack before facing Joachim Low - his assistant when he managed Germany at the 2006 World Cup - and underlined the importance of a squad ethic if Germany are to go deep into the competition.
"We needed a bit of freshness there," he explained. "That's why Brad Davis came in for Alejandro Bedoya and Omar (Gonzalez, who replaced Geoff Cameron) got his chance from the beginning.
"We need all 23 in our squad to get through the tournament.
"This is a huge, huge step and now we can't wait for the round of 16."
The USA can expect to face Belgium in Salvador on Monday, providing Marc Wilmots' side are able to close out top spot in Group H when they face South Korea in Sao Paulo.
