They will take on the defending champions Real Madrid in the final in Kiev on 26 May.
Liverpool boss Jurgen Klopp, who steered Borussia Dortmund to the 2013 Champions League final, described the second leg as "crazy".
That the madness descended was a testament to Roma's ferocious spirit, Liverpool's resilience and contentious officiating.
Monchi, Roma's sporting director, claimed that the Champions League needed the video assistant referee (VAR) system after what he branded incredible errors.
"Congratulations to Liverpool but it is necessary to analyse what happened," he told Italian television. "There were two very clear penalties that were missed and that changed everything. In the first leg they scored an offside goal.
"The VAR must come into the Champions League. I don't understand why there is no VAR in the most important competition.
"Juventus were also eliminated because of incredible mistakes. There were also incredible mistakes during Real Madrid's semi final against Bayern Munich."
Despite Monchi's rant, the visitors built on their first leg victory within 10 minutes through Roman decadence.
Radja Nainggolan lingered on the ball and lost it in midfield. Roberto Firmino swept forward and fed it to an unmarked Sadio Mané who thrashed home past the exposed Roma goalkeeper Alisson.
Six minutes later Roma were level. Liverpool defender Dejan Lovren's attempt to clear the ball crashed into James Milner's head and past the Liverpool goalkeeper Loris Karius.
That made it 6-3 on aggregate. It was 7-3 after 25 minutes when Georginio Wijnaldum nodded in.
That left Roma needing four unanswered goals to force extra time.
Inspired by their three goal comeback against Barcelona in the second leg of the quarter-final, the hosts poured forward.
Edin Dzeko scoed to make it 2-2 seven minutes into the second-half. But despite roars of encouragement from 6O,000 fans, Roma could not find the breakthroughs until the closing stages when Nainggolan struck in the 86th minute and the third minute of stoppage time.
"It's the first time we weren't realy as good as we can be," said Klopp. "We needed luck and that is what we got. 7-6 on aggregate sounds crazy because it is crazy. We came into the competition as qualifiers and now we're in the final and will go to Kiev.
"Roma are strong. We had them where we wanted as they were wild. They ran the risks and we went 2-1 up. All good.
"But then we didn't do enough on the counter. We didn't use the space that they gave us and we lost the game after going 1-0 up."
Liverpool have won the competition in its guises as the European Cup or Champions League five times. They will be attempting to halt the 12 time winners Real Madrid from lifting the trophy for a third consectuive year.
"We will have to play much better in the final than we did against Roma," Klopp conceded. "The good news is we can play much better."
Di Francesco hailed his players following the second leg victory.
"We have grown through this experience but I have regrets because this team should have believed more in its chances, as I did," he said.
"We could have had more goals and pushed Liverpool even more."