Luiz Felipe Scolari's side booked their place in the quarter-finals of the tournament as they edged their opponents 3-2 on penalties in Belo Horizonte, as Gonzalo Jara struck the post with the fifth spot kick for Jorge Sampaoli's men.
"Chile have been one of the great teams of this World Cup without a shadow of a doubt," Alves told FIFA.com. "Today luck was by our side and not theirs, but they deserve all the praise imaginable.
"The way they are able to put pressure on the opposition defence is simply amazing. We knew this was going to be one of our toughest tests and, indeed, it is a gigantic barrier we have overcome."
Brazil now face Colombia in the last eight of the competition next week.