Erik Lamela's 20th minute tap in after a miraculous Zeljko Brkic save from Alessandro Florenzi's close-range header was cancelled out by Luis Muriel's smart individual goal just after the hour mark.
Udinese defender Thomas Heurtaux was sent off in the 73rd minute for a late, high challenge on Florenzi but Roma failed to capitalise on their one-man advantage.
Roma are now on 44 points and remain seventh, three points behind fifth-placed Lazio who occupy the second Europa League spot, while Udinese stayed ninth on 41 points.
Lazio will be looking to close the gap behind third-placed AC Milan to one point when they host fellow challengers for a Champions League place Fiorentina on Sunday (1945 GMT).
League leaders Juventus take on this season's revelation Catania (1400), who are eighth, while second-placed Napoli travel to Chievo Verona at the same time.