The Hoops got off to a perfect start when midfielder Ki Sung-yueng scored from the spot in the third minute after striker Gary Hooper had been fouled by Joel Ekstrand.
The game ebbed and flowed, especially in the second half, and it looked like Neil Lennon's men would see the game out.
However, in the 86th minute Hooper, back defending, gave away a penalty when he clipped Neuton in the box and Abdi sent Lukasz Zaluska the wrong way from the spot.
It was hard luck for Celtic who have one point from two games following their 2-0 defeat by Atletico Madrid in Spain earlier in the month.
Lennon sprung some more surprises with his team selection.
Keeper Fraser Forster was surprisingly replaced by Zaluska, while stopper Glenn Loovens and striker Anthony Stokes were also dropped.
With left-back Badr El-Kaddouri cup-tied, in came Charlie Mulgrew, Victor Wanyama and Mohamed Bangura, who was making his first start.
There was even less familiarity about the Serie A side's line-up.
Udinese were without veteran star Antonio Di Natale, who was allowed to rest back in Italy and coach Francesco Guidolin was questioned for his decision not to bring any of his other recognised first-team strikers to Glasgow.
The visitors, indeed, probably had only four players who could be called regulars - keeper and captain Samir Handanovic, Dusan Basta, Danilo and Emmanuel Badu - in their starting XI and they were looking for goals from Diego Fabbrini and midfielder Almen Abdi.
Celtic had lost only once at home before to Italian opposition over 90 minutes which gave the home fans some encouragement, albeit live television and the early kick-off had clearly affected the attendance, with Parkhead less than half-full at kick-off.
In his pre-match press conference Lennon had warned his side against losing another early goal in Europe.
However, it was the Hoops who took the early lead after when Hooper was clumsily pulled down by Ekstrand.
Referee Bulent Yildirim did not point to the spot at first but after appearing to wait for some guidance from his linesman he eventually confirmed what had looked obvious.
Spot-kicks had been an Achilles heel of Celtic recently but after some delay Ki confidently slammed the penalty high past Handanovic.
In the 22nd minute Zaluska pushed a long-range shot from Badu round the post but the visitors could not capitalise on the corner,
Moments later Celtic's stand-in skipper Beram Kayal was booked for a foul on Badu.
Just on the half-hour mark Hooper forced Handanovic into a fine save with a powerful drive from the edge of the box after being set up by James Forrest but it was mostly all huffing and puffing until the break.
Right-back Mark Wilson replaced Joe Ledley at the interval with Adam Matthews moving to the other side of the park while, for the visitors, Mauricio Isla and Medhi Benatia came on for Roberto Pereyra and Dusan Basta.
In the 51st minute Badu was booked for illegally halting Forrest's progress as he drove at the Udinese defence and from the resultant free-kick, Mulgrew's curled shot was heading for the top corner until Handanovic flew to his right-hand side to parry clear.
Celtic looked in no real danger of losing their lead although a few crosses into their box, as the Italian side tried to battle back, had the fans holding their breath.
In the 62nd minute Wanyama fouled Fabbrini 30 yards from goal and from Cristian Battocchio's chipped free-kick, Abdi had a shot which ended at the feet of Benatia who slipped the ball in to the net from close range.
However, after another delay and some confusion as to why, referee Yildirim awarded Celtic a free-kick inside the box.
Forrest was replaced by Georgios Samaras in the 71st minute and as the game opened up Celtic went looking for the goal which would surely guarantee the three points.
Abdi drove just wide from distance in the 84th minute and moments later, after Samaras led a Celtic counter-attack, Ki's thunderbolt drive was tipped over the bar by Handanovic.
At the other end Zaluska made a fine save from Isla, who only had the keeper to beat, but within seconds Hooper clumsily conceded a penalty for a foul on Neuton and Abdi sent Zaluska the wrong way from the spot.
In the first minute of injury time, after Benatia fouled Samaras, 25 yards out, Mulgrew forced another great save from Handanovic with a powerful free-kick but the game ended even with Celtic fans affording their players a round of applause for their efforts.