Barcelona entered the match holding a 4-1 advantage from the first leg at Camp Nou, but were eliminated from the competition on away goals by an impressive Roma outfit.
A Daniele de Rossi penalty made it 2-0 to the home side in the 58th minute after Edin Dzeko had secured the early breakthrough, before Kostas Manolas headed a third for the Italians in the 82nd minute.
Barcelona had the first half-chance of the match in the fourth minute when Roberto was played into the Roma box, but the midfielder fired straight into the arms of Alisson.
It was Roma that took the lead in the sixth minute, with a wonderful pass from De Rossi releasing Dzeko, and the centre-forward finished past Barcelona goalkeeper Marc-Andre ter Stegen to cut the aggregate score to 4-2.
The home side almost scored their second of the night in the 14th minute when Schick met a whipped corner from the right, but the forward's header just dropped wide of the far post.
The Serie A giants continued to dominate in the period that followed, with Nainggolan and Dzeko both having efforts from distance as Barcelona struggled to make their mark in Rome.
Schick had a golden chance to double Roma's advantage just before the 30-minute mark when he met a cross from Federico Fazio, but the 22-year-old's effort was wide, before Dzeko tested Ter Stegen with a towering header seven minutes later.
Lionel Messi had the opportunity to threaten from a free kick late in the half, but the Argentine missed the crossbar with an attempt that summed up Barcelona's first 45 minutes.
Roma deservedly scored their second of the night in the 58th minute, and it came from a De Rossi penalty, with the Italian finding the back of the net after Dzeko had been brought to the ground by Gerard Pique.
Barcelona were not at the races in the first period, and that remained the case early in the second as Roma continued to force the issue as they looked for the third goal that would have taken them through to the semi-finals on away goals.
Kevin Strootman had a half-chance for the home side 25 minutes from time when he found space outside the Barcelona box, but the Netherlands international missed the crossbar from distance, before Nainggolan tested Ter Stegen from close range.
De Rossi then headed just wide of the Barcelona post in the 69th minute as the intense pressure continued.
Cengiz Under was next to try his luck for the home side as he curled one over the crossbar in the 78th minute with Barcelona struggling to deal with the one-way traffic, before Ter Stegen did well to keep out a close-range Stephan El Shaarawy effort.
Roma spectacularly made it 3-0 in the 82nd minute, however, when Manolas headed an Under corner into the back of the net to send the home side ahead on away goals.
Ousmane Dembele had a cross cleared just over the home side's crossbar in the period that followed, before Messi was denied by Alisson after working his way into the Roma box.
Dembele then fired over the crossbar late on as Barcelona looked for the vital goal, but Roma held on to secure their spot in the final four in incredible fashion.
ROMA (3-5-2): Alisson; Jesus, Manolas, Fazio; Florenzi, Nainggolan (El Shaarawy 77'), De Rossi, Strootman, Florenzi; Dzeko, Schick (Under 73')
BARCELONA (4-4-2): Ter Stegen; Semedo (Dembele 85'), Pique, Umtiti, Alba; Roberto, Busquets (Alcacer 85'), Rakitic, Iniesta (Gomes 81'); Messi, Suarez