After suffering a heavy 4-0 first-leg defeat in the Italian capital last week, the Seagulls came out on top by a 1-0 scoreline in the reverse fixture at the American Express Stadium on Thursday night.
Danny Welbeck scored the only goal of the second leg with a superb long-range strike towards the end of the first half. However, for all their dominance in possession, Brighton ultimately lacked the cutting edge required to pull off a miraculous comeback against a well-drilled Roma outfit, who managed the second leg well to secure their spot in the quarter-finals.
In front of a lively home crowd in full voice, Roberto De Zerbi's side made an energetic start to the contest, but a misplaced pass from Simon Adingra presented Roma with the game's first decent chance in the opening 15 minutes.
Adingra carelessly sprayed the ball across Brighton's penalty area and Sardar Azmoun pounced on the loose ball before firing a venomous strike narrowly wide of the far post.
Brighton continued to dominant proceedings in the final third; Billy Gilmour lacked composure when he blazed a volley over the crossbar from around 12 yards out not long after Pascal Gross dragged a right-footed shot wide from close range.
Azmoun then thought he has scored an outstanding acrobatic effort with his back to goal that looped over Seagulls goalkeeper Bart Verbruggen, but the goal was surprisingly disallowed for an apparent high boot on Jan Paul van Hecke, must to the anger of the Roma players and head coach Daniele De Rossi on the touchline.
Despite Roma's best efforts to break up the play, all the momentum was with Brighton and they eventually broke the deadlock in the 37th minute though Welbeck, who found space on edge of the box and shifted the ball onto his favoured right foot before bending a delightful 20-yard strike into the top corner.
Tempers briefly flared during a set-piece towards the end of the first half after Van Hecke bizarrely decided to touch heads with Gianluca Mancini, who threw himself to the ground to spark a reaction from both sides - VAR decided not to intervene, though.
Following a subdued start to the second half, Ansu Fati and Facundo Buonanotte were both brought on to provide a fresh spark in the final third, and the former almost picked out Welbeck with an inviting delivery into the six-yard box, but Roma goalkeeper Mile Svilar was alert to punch the ball clear.
Not long after Roma's left-back Leonardo Spinazzola struck the woodwork from a tight angle following a wonderful solo run down the left flank, Svilar at the other end made a crucial save from point-blank range to keep out a close-range header from Adingra, who met a perfectly-weighted delivery from Gross.
A towering header from Van Hecke sailed over before Svilar parried a long-ranger effort from Igor Julio straight to Adingra, but he could not sort his feet out and eventually sliced a strike over. Fati then unleashed an audacious 35-yard strike that dipped towards Svilar, who did well to tip the shot over the crossbar.
The efforts of De Zerbi's men in the final third were ultimately in vain, as Roma defended well in the closing stages to see out a comfortable aggregate victory and strengthen their hopes of going one better in the Europa League this season after losing in the 2022-23 final.
Brighton can be proud of their debut campaign in Europe, though, winning five of their eight matches and finishing top of a challenging group ahead of Marseille, Ajax, and AEK Athens. However, a last-16 clash with Roma proved to be a step too far for Seagulls.
Roma, who will discover their quarter-final opponents in Friday's draw, now turn their attention to Sunday's Serie A home encounter with Sassuolo, while Brighton are next in action after the international break when they travel to Liverpool in the Premier League on March 31.