Shunsuke Nakamura saved Celtic's blushes with an 87th minute strike to see off the challenge of Livingston.
The visitors survived a few scares before Shaun Maloney put Celtic ahead from the penalty spot after he had been tripped by Greg Strong.
But the SPL leaders were stunned when Paul Dalglish drove in a shot after 57 minutes after a terrible defensive slip from Bobo Balde.
Celtic's late pressure was rewarded when Nakamura fired in a great winner.
There was a delay of approximately 45 minutes due to a power failure which left the crowd a bit flat.
However, the mood inside Celtic Park was lifted when new signing Roy Keane made an appearance on the pitch minutes before kick-off.
Celtic started brightly and John Hartson mis-kicked from close range when Maloney set him up with a cut-back from the byline.
Livingston striker Dalglish, son of Celtic legend Kenny Dalglish, came close in the ninth minute with a tremendous run before Artur Boruc pulled off a fine save from the angled shot.
Then Livi keeper Roddy McKenzie fortuitously saved a close-range drive from Maloney with his left foot before the Celtic striker missed with a header from the rebound.
The home side continued to press and on the half hour mark McKenzie again saved a Maloney shot from close range with his feet.
However, in the 38th minute the Hoops eventually went ahead.
Strong recklessly pulled down Maloney in the penalty area and the diminutive striker picked himself up to lash in the spot-kick.
Maloney gave Celtic a first-half lead
Moments before the interval Maloney tested McKenzie with a powerful shot but the keeper parried clear.
In the 59th minute Livingston were gifted the equaliser when the home side contrived to lose a goal out of the blue.
Balde tried to walk the ball out of his own penalty area and was robbed by Almondvale striker Robert Snodgrass.
The ball broke to the lurking Dalglish and the Livi striker found the net with a powerful drive from the edge of the box.
In the 74th minute Celtic brought on debutant Michael McGlinchey for Aiden McGeady but the youngster missed a glorious chance to become an instant hero four minutes later, firing wide from close range.
Celtic continued to press against the tiring visitors and they finally got their reward.
Nakamura picked up a Paul Telfer pass, jinked past a couple of tackles then fired in a tremendous shot from 18 yards.
Celtic: Boruc, Telfer, Balde, McManus, Wallace (Virgo 85), Nakamura, Petrov, Lennon, McGeady (McGlinchey 73), Maloney, Hartson. Subs Not Used: Marshall, Thompson, Du, Camara, Pearson.
Goals: Maloney 39 pen, Nakamura 87.
Livingston: McKenzie, Vincze, Strong, Mackay, Tierney, Walker (McPake 83), Brittain, Pinxten, Dair, Dalglish (Dorado 87), Snodgrass (Scott 83). Subs Not Used: Tesevic, Adam, Roy, Boyd.
Goals: Dalglish 58.
Ref: J Underhill