The Foxes scored twice in the final 13 minutes in Eindhoven to reach the Europa Conference League semi-finals
James Maddison and Ricardo Pereira scored in the final 13 minutes in Eindhoven to send Leicester through 2-1 on the night and on aggregate.
The Foxes hit back in their quarter-final second leg after going behind to Eran Zahavi’s first-half opener and will face Bodo/Glimt or Roma in the last four.
It is the first time in the Foxes’ history they have reached the semi-final of a European competition.
Rodgers said: “We’re very young in our experience as a club in Europe, but nights like this, everyone grows from it – the players, the club, the supporters – and when you get to your first semi-final it’s a great feeling.
“The boys were absolutely outstanding in the second half. When you come to finals, semi-finals and big games you don’t get a second chance.
“They kept going and kept fighting and we scored two wonderful goals. I go into games having a script how it can go, I map out different things which may happen, and I felt we needed to bring the intensity in the second half.”
After last week’s 0-0 first-leg draw in Leicester, both teams took off the shackles and Kasper Schmeichel brilliantly saved Mario Gotze’s volley after 14 分钟.
Two minutes later Harvey Barnes shot wide when through and PSV took advantage by grabbing the lead midway into the half.
Youri Tielemans’ sloppy pass was seized on by Gotze and he slipped in Zahavi to drill an angled effort low past Schmeichel.
Maddison nearly found an instant leveller only for Jordan Teze to clear his deflected effort off the line and the chances continued for Leicester after the break, with Rodgers making two key changes by bringing on Ademola Lookman and Patson Daka.
PSV initially looked stronger but faded badly and Daka should have equalised just after the hour instead of shooting wide when sent clear by Lookman.
Leicester continued to press and eventually found a leveller with 13 minutes left. Ayoze Perez bustled his way to the byline and crossed for Maddison to sweep in his 14th goal of the season.
The Foxes were on top and grabbed a dramatic late winner with two minutes left when Yvon Mvogo parried Daka’s shot and the ball fell for Pereira to score.
Schmeichel added: “We don’t make it easy for ourselves, but it goes to make a better story. I’m just very, very happy and proud of the boys.
“We are a young club when it comes to European football and we were up against a juggernaut of European football. To come away from home and make it difficult, have that luck go against you and get the result, that’s a sign of top mentality and kind of thing we strive for.”
Defeat for PSV extended their 17-year wait for a European semi-final and is a blow ahead of Sunday’s Dutch Cup final against Ajax.
Boss Roger Schmidt said: “It’s always the same in football, the goals were decisive. We scored a good goal, had moments to score the second goal and you cannot prevent everything.
“For now I’m going to be disappointed and on Friday when I open my eyes I will think about Sunday.”