The Giallorossi had trailed 4-1 after the first leg, but they blew LaLiga’s leading side away at the Stadio Olimpico as goals from Edin Dzeko, Daniele De Rossi and Kostas Manolas clinched a dramatic triumph.
Dzeko was sublime throughout and netted early before winning a penalty after the break for De Rossi to pull Roma to within one goal of the last four.
An out-of-sorts Barca still looked as though they would do enough to scrape through as chances came and went for the hosts, but Manolas – who, like De Rossi, had put through his own net at Camp Nou – headed home eight minutes from time to put Eusebio Di Francesco’s side in the driving seat.
Although the visitors suddenly mustered a challenge, they could not force a decisive goal and Lionel Messi and co crashed out in scarcely believable circumstances.
Roma last reached the last four of Europe’s premier club competition in 1983-84 when they lost to Liverpool in the final, and they will now fancy their chances against anyone – including the Reds and former star Mohamed Salah, who defeated Manchester City in Tuesday’s other quarter-final.