In today's other match Mohamed Salah's remarkable run of goals continued to help send Liverpool into the last four as Jurgen Klopp's side came from behind for a 2-1 win at Premier League rivals Manchester City and a 5-1 aggregate triumph.
"They showed attitude and for whatever reason we didn't know how to respond, or to make chances", Barcelona midfielder Sergio Busquets said.
Apart from Edin Dzeko, Daniele De Rossi and Kostas Manolas were on the scoresheet for Roma as a hapless Barcelona failed to register a single goal.
Di Francesco acknowledged before the match that his team needed a "miracle" to advance, and they got the start they needed when when Edin Dzeko controlled an over-the-top pass between two defenders after six minutes and poked in his sixth goal in this season's competition.
It was an incredible night at the UEFA Champions League quarter-finals after Barcelona and Manchester City - two clubs deemed favourites to lift the title due to their league dominance - were eliminated in the quarter-finals stage - by AS Roma and Liverpool - leaving the entire football fraternity baffled with the results.
Barcelona had one foot in the semi-finals after winning the first leg 4-1. "We had a great opportunity and we didn't seize it".
"I'm not sure you cannot concede when you've got two strikers that don't put anything into defending", he said, per MailOnline's Robert Summerscales.More news: Australian media report China proposes military base in South Pacific
The victory took the Italians into the European Cup semi-finals for the first time in 34 years.
Coach Eusebio Di Francesco is taking the plaudits for a tactical masterstroke which saw Roma play with a three-man defence for the first time under his guidance, and with two centre forwards, with Dzeko partnered by Patrik Schick.
Roma had to wait until the 58th minute for their next goal but Dzeko was a central figure again, holding up the ball before being hauled to the ground by defender Gerard Pique. "If you come all this way and get to the semi-finals, why shouldn't you believe in getting to the final in Kiev", he asked reporters after the historic win.
"But we made an outstanding 45 minutes, really good. We knew the gap between us and them wasn't 4-1".
Di Francesco said the aim of the change in formation was to make his side more unsafe down the flanks and also praised his players for their near-perfect display.
"I take the blame for defeats and some ungrateful words on social media, so I may as well take the credit for a win", he told Mediaset Premium.