Atleti looked to have a realistic route to the final after being put in the same half of the draw as Leipzig, Paris Saint-Germain and Atalanta.
But after Tyler Adams struck in the 88th minute to restore Leipzig's lead following Joao Felix's response to a Dani Olmo opener, the Bundesliga outfit advanced in Lisbon to earn a showdown with PSG.
Atletico midfielder Saul acknowledged the scale of the disappointment for the club's supporters in Lisbon on Thursday.
"The opponents were superior in many phases of the game," the Spain international said post-game.
"We were very excited. The fans deserved more. We had a point of hope and we are very sorry."
Meanwhile, captain and team-mate Koke voiced his disappointment at failing to capitalise on the unique format of this season's competition, where one-legged ties bring the latter stages closer into view.
"We were very excited about this Champions League and now it's time to get up," he said. "On the field, sometimes you are better and sometimes worse. We have to see what has failed.
"This is a different competition. In three games, we could have won the Champions League. Now that will not be able to happen."