Erik Lamela's 20th minute tap in for the visitors, after a stunning Zeljko Brkic save from Roma midfielder 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 then sent off in the 73rd minute for a late, high challenge on Florenzi but Roma failed to capitalise on their one-man advantage.
They remain seventh on 44 points, three points behind fifth-placed Lazio who occupy the second Europa League qualifying spot, while Udinese stayed ninth on 41 points.
"It was a good game against a good team like Roma, and in the end it was a decent draw for us," Udinese forward Antonio Di Natale told Sky Sports Italia.
On Sunday, league leaders Juventus host this season's surprise package Catania, who are eighth, while second-placed Napoli travel to Chievo Verona (both 1400 GMT).
In Rome all the talk surrounded the prospect of 36-year-old Francesco Totti taking over from Gunnar Nordahl as the second highest Serie A scorer of all time behind Silvio Piola (274) after he drew level with the Swede on 225 goals last weekend.
But it was the 21-year-old Lamela, Roma's leading scorer this season, who put the hosts in front.
Florenzi had already spurned one golden chance to score when Totti again picked him out with a beautiful cross, only for the midfielder to head straight at Brkic.
The keeper tipped the ball on to the bar but it dropped straight at the feet of the onrushing Lamela, who tapped his 12th goal of the season into an empty net.
Udinese were a different side in the second period and got the goal their pressure deserved after 62 minutes through Muriel, another 21-year-old attacking talent.
The Colombian striker cut in from the byeline and skipped past Nicolas Burdisso and Vassilis Torosidis before firing the equaliser through Maarten Stekelenburg's legs.
Udinese were reduced to 10 men 11 minutes later when referee Marco Guida showed Heurtaux a straight red card for his clumsy tackle on Florenzi, infuriating his team mates, who surrounded the official in protest.
"I spoke to Thomas about the sending off and he said that yeah he caught him, but it wasn't such a bad tackle as to deserve a red card," said Muriel. "But it's the referee's decision and we can't do anything about it."
Roma never really looked like taking advantage of having the extra man with Udinese more than holding their own, although Pablo Osvaldo could have snatched the points for Aurelio Andreazolli's side in injury time.
The Italy striker found himself in front of Brkic without a defender near him but shinned his attempt to lob the stranded keeper and watched his shot drifted harmlessly wide leaving the game to finish level which neither side will be happy with.
On Sunday, Roma's local rivals Lazio will aim to close the gap with third-placed AC Milan to one point when they host Fiorentina (1945 GMT) who are also vying for Champions League football next season. Milan won 2-0 at lowly Genoa on Friday.