Rodrigo Taddei and substitute Mirko Vucinic scored Roma's goals, which sandwiched an equaliser struck by Real skipper Raul.
"It is certainly a very satisfying experience. The lads played a perfect game and must now enjoy this joy we are all sharing," smiled Spalletti.
"It was a great performance and we had to suffer a little after the equaliser when Real had nothing to lose, but overall we utterly deserved to qualify.
"The side showed maturity. Those who played and those who came off the bench did very well and took the right approach, transmitting enthusiasm and character to their teammates. With a mental attitude like that we can get results anywhere."
"If you see those who came off the bench made a crucial contribution, then in terms of the collective we have more advantages to having players of quality in all areas.
"They also helped their teammates from the sidelines with encouragement and comments that you need in big games like this."
Real Madrid barely had a shot on goal, even before Pepe's 70th-minute dismissal, despite losing the first leg 2-1 in Rome.
"We did well to lower the centre of the squad in some situations, but when needed we nipped at them so they could not pass the ball around too comfortably in our area. Our strikers often went to push them and speed up the tempo so they weren't comfortable on the ball.
"I think Real's performance tonight depended on our own approach. I probably did expect a little more danger from them, but that was down to us not allowing them any space."
