Shakhtar were defending their lead from the first leg, where they had triumphed 2-1, but Roma's Edin Dzeko scored an early second-goal to crush their European dreams.
Speaking after the victoy, which marks the first time that Roma have reached the quarter-final stages of the Champions League since 2008, Di Francesco said: "This is an extraordinary result because it's not easy to play at these levels. This was a team made up of men who I won't say played the game of their lives, but almost.''
He added: ''Roma have had the satisfaction this year, despite some difficult times, of getting into the quarter-finals of the UEFA Champions League and we need to be proud of this. And as for our fans - I had goosebumps. [Now] I don't care who we play [in the quarter-finals]. We need to let this sink in.''
Di Francesco is quietly confident that Roma's strong form thus far will lead them to victory, arguing: ''We've won five of our last six in the league and we're into the last eight in Europe, so I am seeing growth in this team. I am totally convinced that this team is going places.''
The only team the Italian would not like to face, though, is Sevilla as he targets a showdown with former team-mate and colleague Vincenzo Montella in the final on May 26.