Tore Andre Flo's 60th-minute goal was sending Chelsea through despite Rivaldo and Luis Figo scoring before the break, but Dani Garcia scored seven minutes from the end of regulation to force extra-time.
Roma had begun the comeback early when Edin Dzeko controlled an over-the-top pass between two defenders and poked in after six minutes for his sixth goal in this season's competition.
The Serie A side trailed 4-1 to Barcelona from the first leg at the Nou Camp, but thrashed the LaLiga leaders 3-0 on their home patch.
And following the match, Di Francesco said it was his tactical switch to a back three that set his team on their way to the historic win.
He sent two glorious headed chances wide, the first from a vicious 13th-minute Aleksandar Kolarov corner.
Barcelona looked rattled as Roma pressed with high intensity and threw everything at them when in possession.
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Roma started the second half as they started the first and got their reward in the 58th minute when Edin Dzeko was chopped down by Gerard Pique and De Rossi slotted home the spot-kick.
They do have an away goal in their advantage, but they need to score three times while keeping Barcelona silent for that to count for anything.
The Italians managed to carve out another chance in the 69th minute but Ter Stegen was on hand to keep out Kevin Strootman's effort from distance.
And after substitute Stephan El Shaarawy had been denied by a last-ditch save by Ter Stegen at the back post in the 79th minute, Manolas nodded Cengiz Under's cross past the German to make it 4-4 on aggregate with eight minutes left to play.
While opponents have failed to hurt either team domestically, they have found life tougher in the Champions League.
Alongside Manchester City, Barcelona were one of the favourite to win this year's Champions League, but we will see semi-finals featuring Liverpool and Roma instead.