James Maddison and Ricardo Pereira scored the decisive goals in the final 13 minutes to send Leicester through 2-1 on aggregate.
The Foxes had to show resolve after conceding in the first half to Eran Zahavi and they will now play in their first semi-final of a European competition in its history.
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 stalemate at the King Power Stadium, both teams looked to open proceedings and it started with Kasper Schmeichel making a brilliant save from Mario Gotze's volley after 14 minutes.
Two minutes later Harvey Barnes missed the target and PSV made them pay by taking the lead.
Gotze took advantage of Youri Tielemans' sloppy pass and he assisted Zahavi who scored past Schmeichel.
Maddison was denied an equaliser straight after and chances continued for Leicester in the second half. Rodgers made two substitutions and brought on Ademola Lookman and Patson Daka and they made an instant impact.
Ayoze Perez crossed for Maddison to sweep in his 14th goal of the season to level the game with just 13 minutes left.
And with two minutes of normal time remaining, Leicester scored a dramatic winner when Daka's shot rebounded off the keeper for Pereira to score.
Schmeichel said: "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."
They will now play Jose Mourinho's Roma in the last four of the competition after the Italian side overcame FK Bodo Glimt 4-0 in the second leg.