Jose Mourinho's side brought a one-goal lead into the second leg in Germany, and the result also stood at full-time on Thursday night, with the Serie A giants progressing to the final of the competition.
Roma, who triumphed in the Europa Conference League last season, are looking to win this competition for the first time, and they will take on either Sevilla or Juventus in the final on May 31 in Budapest.
Lorenzo Pellegrini had the game's first shot in the second minute, with the Roma captain collecting a pass from Tammy Abraham before drilling one wide of Lukas Hradecky's post.
Kerem Demirbay and Moussa Diaby then took it in turns to have attempts for Leverkusen, with the former firing an effort straight at Roma goalkeeper Rui Patricio, before the latter turned wide of the post after smart work from Florian Wirtz in an advanced area.
Diaby then hit the crossbar with a rasping effort in the 12th minute, before Patricio did well to keep out a long-range effort from Demirbay, with Leverkusen looking the more likely to make the breakthrough.
Patricio was again called into action in the 27th minute to deny Demirbay, while Diaby also headed wide of the post, with the two forwards causing Roma a lot of problems.
Patricio had to make another save to deny Sardar Azmoun in the 36th minute, and then Demirbay curled one wide of the target in the final chance of the first period, with Roma still leading 1-0 in the tie despite struggling for long spells in the opening half of action.
The match was temporarily stopped at the start of the second period due to smoke from flares, which had spread across the pitch.
Patricio was again involved in the 66th minute, the Roma goalkeeper keeping out a header from Azmoun, with the Italian giants still ahead in the tie heading towards the final 20 minutes.
Excellent defending from Gianluca Mancini denied Azmoun a clear sight of the Roma goal in the 67th minute after Patricio had palmed a strike from Demirbay into a dangerous area.
Azmoun continued to cause problems and dragged another effort wide of the post in the 81st minute, but there was no way through for Leverkusen on a frustrating night for Xabi Alonso's side, who had more than 70% of the possession and 23 shots to Roma's one.
Eight additional minutes were added in Germany, but Roma continued to defend well, and Mourinho will now be looking to win this competition for the third time later this month.
Leverkusen will now switch their attention back to the Bundesliga, preparing to host Borussia Monchengladbach on Sunday, while Roma will continue their Serie A campaign against Salernitana on Monday.