Goalkeeper Hart, who was named captain on what could be his final City appearance, received vocal backing from the home supporters amid speculation over his future under new boss Pep Guardiola.
Fabian Delph headed the only goal on the night at the Etihad Stadium as City completed a 6-0 aggregate win over the Romanians.
When asked about his future, Hart told BT Sport: “That’s not for me to say; it is what it is.
“That was a really special night for me. I’ve had a lot of good nights in football.
“We all know there’s a situation going on but I feel that inside, outside of all the circus, we are handling it well as a good team with a good management staff.
“This is a special place for me, there’s never been any secret made of that. It’s the place I’d love to be but situations occur in football but we’re men and we get on with it.
“Tonight was really nice and I really appreciate the people at Manchester City and it turns out they appreciate me.
“It’s a situation, we’ve got a top manager that the club’s wanted for a long, long time and he’s going to have his opinion on things.
“Outside of the charade and all the rubbish that is talked we are dealing with it like men and we are going to come up with a solution.”
Hart’s ability with the ball at his feet has been cited as a possible reason for any departure, but when told he had a 100 per cent pass completion rate on the night, he joked: “Pigs do fly!”
Guardiola divulged little regarding Hart’s future, although he did admit that as the manager of Manchester City it is his prerogative to make decisions on the make-up of his team.
The Spaniard said: “Joe is still a player of us. I am so happy with what he did.
“I know he is a legend for this club. I know part of what the club has become – one of the reasons is Joe. They won the Premier League, the cup. He helped to make this team one step above. I am so happy how the people treat him and love him.
“But I am here to take decisions. I make a lot of mistakes – my decisions as a coach are sometimes good, sometimes bad, but I take decisions.
“I try to be honest with myself. I can’t deny what I feel. I was honest with Joe, the club and all the players in the squad. The people must follow me.”