The visitors, needing victory in the final fixture to go through into the knockout stages, deservedly took the lead in the 29th minute when Gary Hooper took advantage of a mix-up in the Udinese defence to tap in from a yard out.
However, veteran striker Antonio Di Natale notched from close range himself with seconds remaining in the first half after Celtic defender Cha Du-ri inadvertently headed into his path.
The second half was drama-filled with Neil Lennon's side playing their part in an enthralling game but in the end, the draw was not enough.
Celtic travel to St Johnstone on Sunday looking to close the gap on SPL leaders Rangers and they will be buoyed by their spirited performance against a side who are only behind Juventus on goal difference at the top of Serie A.
Celtic skipper Scott Brown started for the first time since recovering from an ankle injury picked up against Rangers in September.
With Glenn Loovens ruled out with a hamstring injury picked up against Hearts on Saturday, Victor Wanyama was moved back into defence with midfielder Ki Sung-yueng replacing striker Anthony Stokes who started on the bench.
Hooper and Georgios Samaras started in attack supported by James Forrest while Udinese boss Francesco Guidolin fielded a strong side with top scorer Di Natale leading the line.
The small band of travelling supporters certainly made themselves heard but in the week that they had been fined £12,700 by UEFA after the Hoops fans were reported for "illicit chanting" in a previous Europa League fixture against Rennes at Parkhead, it perhaps was not the best time to unfurl a banner which was derogatory in its sentiments towards European football's governing body.
Nor was the throwing of a flare onto the running track by the Hoops fans too clever.
When the game started Celtic more than held their own in an impressive first-half display, inspired by Samaras and Forrest.
In the 12th minute, though, keeper Fraser Forster came out rather unconvincingly from his goal when his defence was breached by Pablo Armero and was happy to see Di Natale's lob over him just go past the post.
And Lennon must have held his head in hands in the 23rd minute when Hooper missed a great chance.
Samaras used his strength again before playing the former Scunthorpe striker in with a clever pass at the edge of the box but although Hooper's angled drove from 16 yards beat Samir Handanovic, it went past the far post.
In the 27th minute Forster made up for his earlier hesitancy when he pulled off a fine save from Di Natale's drive from the edge of the box which was heading for the far corner.
Two minutes later Hooper made up for his earlier miss when he gave Celtic the lead, albeit there was a hint of good fortune about the goal.
Samaras was again involved with a cross from the left which Handanovic punched off his own defender, Joel Ekstrand, and the ball fell kindly to the Englishman who tapped into the empty net from a yard out.
It was all going to plan for the Parkhead club but seconds from the interval Cha made a mess of a header when he was challenged by Kwadwo Asamoah, allowing Di Natale to volley past Forster from close range, although the Italian international had looked to be in an offside position when Almen Abdi's cross came into the box.
Udinese appeared more focused and determined and in the 56th minute Hoops defender Daniel Majstorovic made a great blocking tackle on Di Natale's goal-bound shot.
Amid the growing tension, Cha was booked in a case of mistaken identity when it should have been Brown.
Forster made fine saves from Di Natale's drive and substitute Mauricio Isla's back-post header, before he made another decent stop from Asamoah's long-range effort.
As Udinese piled on the pressure, and after Stokes replaced Beram Kayal, Asamoah struck the post from three yards out when it looked as if he must score.
But in the 75th minute Cha struck the post with a shot which Handanovic allowed to then escape his grasp and almost trundle over the line.
More confusion arrived when Cha was booked again, this time for a foul on Armero, but referee Michail Koukoulakis did not send him off, appearing to have rectified his earlier mistake.
Amid almost breathtaking excitement Forster made two great saves from Di Natale before Mo Bangura replaced Samaras.
There was a penalty claim for handball in the Udinese box as Celtic threw everything forward in the final minutes.