Erik Lamela had set Roma on course for a fourth successive Serie A win but Luis Muriel hit back in the second half and though Thomas Heurtaux saw red soon after, the capital club failed to take advantage.
Roma remain seventh, three points behind a fifth-placed Lazio side that occupies the second Europa League spot, while Udinese stay ninth on 41 points.
Lamela was afforded the simplest of tasks when Zeljko Brkic could only parry Alessandro Florenzi's close-range header.
But Muriel delivered a fine solo riposte, freeing himself from two defenders before shooting under the legs of Roma goalkeeper Maarten Stekelenburg.
The momentum seemed back with the visitors when Heurtaux was sent off (73) for a late, high challenge on Florenzi but a winner proved elusive.
Lazio can cut the gap on third-placed AC Milan to one point when they host Fiorentina on Sunday.
League leaders Juventus take on surprise package Catania, while second-placed Napoli travel to Chievo Verona.