The Nerazzurri seemed done for after losing the first leg 3-0 at White Hart Lane, but a 3-0 victory forced extra time where they eventually went out on away goals, 4-4 on aggregate.
“It was an important performance tonight,” Stramaccioni told Sky Sport Italia.
“There was a strange atmosphere and so much criticism… We always give our response on the field. We made a huge step forward in terms of our football too, finding certainties that we haven’t always had. We are all united and want to end this season on a high.”
Stramaccioni even made fun of the rumours he is due to be replaced by Mircea Lucescu.
“Massimo Moratti called me yesterday and said: ‘Hello, Lucescu?’ I replied ‘No, it’s still the other guy.’ The President is fantastic.
“This is a fundamental sign we send today and the only positive I can see from this evening is that we won’t have midweek games to worry about. It was our best performance of the season.”
Inter looked like a completely different team from the one that lost so badly in London only a week ago.
“We had suffered a lot with Moussa Dembele and Scott Parker, whereas today we decided to leave more space to the full-backs. They were more worried and didn’t take advantage of the extra room.
“The team shook off its fear and always held the initiative. It’s a shame about the result, as we conceded on a distraction after so many scoring opportunities.”
Stramaccioni was asked whether this amazing comeback can give Inter the psychological boost they need against Sampdoria on Sunday evening or represent extra fatigue.
“Playing 120 minutes at an incredible level of intensity is tough and some players ended in precarious conditions, but I saw a team that made big steps forward today.
“The important thing is that today Inter gave their response, showed that we are united and that we won’t be broken by the criticism.”