The Barcelona arrived in Spain’s capital, at the Vicente Calderon with a 2-1 advantage from the first leg but poor defending opened their misfortunes as the unmarked Antoine Griezmann punished them.
Saul headed against the bar as Barca toiled before Griezmann scored a late penalty after Andres Iniesta handled.
The holders were beaten 3-2 on aggregate by Spanish rivals Atletico Madrid. Barca was denied a penalty when Gabi handled just inside the area.
It is only the third time in 11 years that Barcelona has not reached the semi-finals of the Champions League.
They are three points clear of Atletico at the top of La Liga, but have lost three of their past four games in all competitions.
“It was the game we predicted, although perhaps I didn’t expect Atletico to play so deep. It wasn’t our best day,” Luis Enrique, the barca coach said, who won the treble in his first year in charge at Barca.
“We have to stay calm and think that we still have two competitions where we are in the running.
“Everyone can choose the explanation. To give mine I have to analyze the performance. It is not our best time, but it is not the time to be nervous.”