Erik Lamela put Roma in front after 20 minutes and the Giallorossi held the upper hand until the 62nd minute when Luis Muriel brought Udinese back on terms.
A red card for Thomas Heurtaux reduced the home team to 10 men for the last 20 minutes of the game, but they still held on for a point which leaves them six points adrift of the top five, while Roma move to within three points of their city rivals Lazio.
Udinese had a good early chance through Antonio Di Natale, but Maarten Stekelenburg did not get caught cold and made sure his first touch of the match was a telling one.
The home team were unable to build on that third-minute attempt, allowing Roma to gain a foothold in the game.
Francesco Totti set up Alessandro Florenzi for a good chance in the 18th minute, but Zeljko Brkic denied him in a one-on-one situation.
And Roma soon took the lead, with Lamela becoming their top scorer for the season with 12 goals.
Florenzi's diving header was pushed on to the crossbar by Brkic, and Lamela was on hand to knock the ball in from close range.
A shot from Maicosuel early in the second half hinted at an improvement from the home team, and earned some rare applause from a sparse crowd inside the Stadio Friuli.
Muriel gave them even more reason to cheer in the 61st minute when he brought Udinese level with his fifth goal of the season, surprising Stekelenburg with a shot which found its way through his legs, just inside his near post.
Udinese's momentum was stalled 10 minutes later when Heurtaux received a red card for dangerous play and, as a consequence, the Zebrette had to abandon their hopes of clinching all three points.
Roma's attempt to take advantage failed to materialise, though, as they came up against a solid defensive wall which held firm.