A late goal from Michael Bradley secured a ninth win in a row for the Giallorossi, with coach Rudi Garcia describing the win as "very important for the future"
The result made it nine wins from nine matches for the league leaders, despite the absence of Francesco Totti and Gervinho through injury, and coach Rudi Garcia was delighted at his side's resilience in a tense afternoon in Udine.
"It's true that winning in this very difficult arena is impressive, as Udinese were unbeaten here in over a year," he told Sky Sports Italia after the match.
"Especially when down to 10 men this victory is very important for us and will be for our future too. We had to remain dangerous and work even harder, playing to win even when a man down, and that is a good sign.
"The goal was wonderful, especially with that move when down to 10 men. More than in other games, today we needed fresh legs off the bench like Bradley and Marquinho."
The victory restores Roma's five-point lead over Napoli and Juventus at the top of the table following wins for the Partenopei and Scudetto holders on Sunday.