The Catalan giants appeared to have one foot in the semi finals of the competition after being 2-0 up at the break, but goals from Cristiano Ronaldo and Karim Benzema put Madrid back in the race until the final whistle.
"We knew we had to suffer until the end," Alves said after the match. "This version of Real Madrid is the one that gives us the most problems, because they create a lot of pressure.
"They were phenomenal, closed down very well and made it very hard for us to attack.
"The most important thing is that we are through. We knew this tie would be decided in both matches, not just the first one.
"We are very happy to go through."
The Brazilian right-back then criticised Madrid's lack of fair play, saying: "A minor detail is that when the ball goes out, we return it and Madrid do not.
"This is the difference between the two teams."
Starting in November and continuing through to the middle of February, Goal.com is publishing exclusive extracts from Graham Hunter's highly-anticipated new book, 'Barca: The Making of the Greatest Team in the World'. Friday's extract looks at how Barca suffered rare disappointment at the hands of Real Madrid in a gripping finale to last season's Copa del Rey. Don't miss it!