The Giallorossi took a second half lead through Andrea Marcucci's penalty, before goalkeeper Stefano Greco saved a Milan spot-kick soon after.
The visiting Rossoneri did equalise soon after that - but Alessio Riccardi lashed home from inside the box with 15 minutes remaining to ultimately secure the win.
"It was a very even game, with both sides looking to win," De Rossi said afterwards. "Milan are one of the few squads who are always looking to push forward a score. We allowed a few chances.
"We still need to improve, we are committing a few too many errors of inexperience. We shouldn't wait for our opponents to attack, but at times I see that we are content to sit back and this isn't our mentality or that of Roma."
See more photos from the game at Tre Fontane in the gallery below.
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