The Giallorossi faced an uphill task heading into the second leg at the Stadio Olimpico after losing 4-1 in Catalonia, but the Serie A side progressed with a 3-0 win after goals from Edin Dzeko, Daniele De Rossi and Kostas Manolas.
Di Francesco opted for a change of formation - deploying a 3-4-2-1 - and he believes this was key to picking up a result, with Radja Nainggolan and Champions League debutant Patrik Schick ably supporting Dzeko.
"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.
"The truth is that I made this choice to create more width, allow more counter-attacks and bring speed, but what really changed was the philosophy of the side and we have to continue like this now.
"Although we have had very good performances in Serie A, we are only now reaping the rewards – the just rewards – for the hard work we have done."
And Di Francesco echoed the thoughts of his captain De Rossi in calling for Roma to keep believing and aim to reach the final of Europe's premier club competition.
"Of course we should believe we can reach the final - why wouldn't we believe?" he said. "We have come this far and nobody expected us to do it, so we should push on and aim for something more, raising the bar every time.
"I don't want a side that rests on its laurels or thinks it doesn't matter how the game goes. I won't stand for that. We have to target more and realise we can always do better."