Edin Dzeko, Daniele De Rossi and Kostas Manolas secured the 4-4 aggregate draw and sent the Stadio Olimpico into raptures, as Barca, who remain unbeaten in LaLiga this season, completely fell to pieces.
They are joined by Liverpool, who beat Man City 5-1 on aggregate in the day's other UCL match, and will await their last-four opponent pending the result of Wednesday's matches with Bayern Munich leading Sevilla 2-1 and Real Madrid holding a 3-0 aggregate lead over Juventus.
"Nobody believed in us and we have reached the semi-final for the first time in the Champions League [era]".
It was a perfectly executed strategy by Roma coach Eusebio Di Francesco, who has taken the Italians to their first European Cup semifinal since 1984.
The second goal came in the 58th minute as Dzeko was pulled down inside the box by Pique and the subsequent penalty was scored by Mr. Roma, De Rossi.
82 MINUTES: "GOAL! Roma 3-0 Barcelona (Manolas) agg 4-4"'ROMA HAVE RISEN FROM THEIR RUINS!' They win a corner down the right, it's brilliantly curled to the near post by Kolarov, and Manolas times his run perfectly, flicking hard and fast and enjoying the celebration like you must!
"We scored two own goals, there were some dubious situations and we knew the gap wasn't 4-1".
"Now we have to take another step - not think we're in the semi-final and have already achieved the maximum". "We so wanted to win this tournament, but we've failed to do so again".
Trailing 4-1 after the first leg, Roma went ahead on the night after six minutes as Edin Dzeko latched on to Daniele de Rossi's long-range pass before poking the ball beyond Marc-Andre ter Stegen. And the pressure continued all the way through the first half, Barca barely mustering an attack - and yet, Ernesto Valverde did nothing at half-time but hope that his team improved.
Suarez was so frustrated at times for not receiving enough balls that he slammed both hands on the pitch on one occasion.
The referee didn't immediately point to the spot but then showed a yellow to Pique and awarded the penalty.
Just as he had done in the World Cup final in 2006, Giallorossi-stalwart De Rossi converted the spot-kick easily and it just started to seem like this would be a night for the home side.
It was De Rossi's first Champions League goal since October 2015.
But three minutes later, the Barcelona goalkeeper had no chance when Manolas headed inside the far post.
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