Leverkusen won 1-0 on the night and 4-1 on aggregate to dispatch a feckless Rangers side that managed just four shots.
They will now face Inter Milan on Aug. 10 in Dusseldorf.
Sevilla, meanwhile, skewered Roma's defense to record a 2-0 victory in a one-off fixture. They were unable to play the first leg due to the coronavirus pandemic.
Julen Lopetegui's men will meet Wolverhampton Wanderers on Aug. 11 in Duisburg.
The Europa League will look different from the quarterfinal stage onward. All games will take place in Germany because of COVID-19 restrictions and the tournament will follow a single-elimination format, up to and including the final on Aug. 21.
Rangers' Europa League campaign ended 395 days after it started. Steven Gerrard's side began its odyssey in Gibraltar against St. Joseph's on July 9, 2019.
Sevilla's hopes of winning a record sixth Europa League title remained intact with a complete performance of their own. On-loan Real Madrid full-back Sergio Reguilon exposed Roma's porous defense in the first half to score the opener, and Youssef En-Nesyri tapped home Lucas Ocampos' crossfield pass to give the Andalusian outfit an unassailable lead. Roma's Gianluca Mancini was sent off in the 98th minute for a needless elbow on substitute Luuk De Jong.