Chivu's side arrive in Madrid full of confidence after fighting back from two goals down to beat Napoli, and face a Real Madrid side already dealing with an opening-week loss to Real Betis.
The Romanian, who won the Champions League as an Inter player under Mourinho at the same stadium in 2010, will go head-to-head with his former coach for the first time as a fellow manager on Tuesday night.
"We have an obligation to write important pages in the club's history. That's what we try to do," Chivu said of the 2010 final and what he wants to see from his current side.
"The stadium has been renovated, it's more beautiful — it already was, but now it's modern too, and we like having the chance to play a match here. Mourinho is a special person, and I was lucky to find him. He helped me through a difficult moment, and I'll never forget that."
Asked whether his own methods carry traces of his old manager's mentality, Chivu was quick to stress his independence.
"I've never copied anyone. I have my own idea, built up over my years as a player, and also with the new generations," he said.
On the scale of the challenge awaiting Inter Milan, who have not won at the Bernabéu in 60 years, Chivu was under no illusions.
"Playing here isn't easy. Many teams have struggled. We'll do everything we can, all that we're capable of. We want to show we're competitive — there's no perfect way to face Real Madrid," he said, adding that his squad depth would matter as much as his starting XI.
"It's not about who starts, but about the 16 players capable of contributing something. We saw that recently in Cagliari and against Napoli. The players from the bench give you energy."
Chivu admitted he would have preferred more recovery time after Saturday's dramatic win but said the calendar left no choice, and was measured when asked where he rates his side among the Champions League favourites.
"We have the ideas, and with the work we're doing, we're in a position where many think we can achieve it. You have to get through moments and have a bit of luck. We want to be competitive and rise to what this competition is," he said.
On facing the current version of Mourinho's Madrid, Chivu was generous in his assessment despite their recent slip.
"It's very early, and beginnings aren't easy for anyone, including new players. It's hard to handle the pressure — playing for this club isn't easy, and I think it's been a balanced start. They didn't deserve to lose because they played well," he said.
"He's very good at immersing himself in the reality of the club he represents and transmitting his mentality. He's a winner — there are few like him in the world. He's already shown that here, but elsewhere too. Real Madrid will have a great season, I'm sure of it."