Late goals from Raheem Sterling and Harry Kane sealed a famous 2-0 win and left Klinsmann claiming a first title since 1966 is there for the taking.
Klinsmann told the BBC: “You’ve got to keep the team right there in the moment, live the day and prepare them in the best way possible for the quarter-finals.
“Don’t go ahead and think, ‘OK, we could be back here for the semi-finals’. But they won’t.
“The energy they had tonight, the chemistry, looks amazing. They all get along, everything seems to be in sync and in balance – it’s your tournament.”
Former England stars Alan Shearer and Rio Ferdinand also believe Gareth Southgate’s men are on the cusp of something special.
Southgate’s men face Sweden or Ukraine in the quarter-finals before a potential semi-final clash with either Spain or Switzerland, who stunned France on Monday night.
Shearer said: “If you could have picked something like that then you would have.
“England will never, ever have a better opportunity of winning the Euros than this opportunity now. If they’re lucky enough to get to the final that will have been six games played at Wembley, and you can’t ask for more than that.
Ferdinand added: “When you look at that draw, I know those lads will be sitting in their hotel thinking, ‘there’s no-one there who we can fear’.
“France were the only team you could look at and say, ‘they’re going to give us a really hard game’. There’s no-one in there you look at and think, ‘we would rather skip them’.