The home side were deservedly ahead at hal- time thanks to a John Hartson header. However the visitors shocked Celtic early in the second half with goals from Guillaume Beuzelin and Steven Fletcher.
Celtic looked stunned at this turn of events, but drew level after a marvellous free-kick by Shaun Maloney before Hartson bundled home the winner after Maloney's shot had come back off the post.
It was breathtaking action and there was still time for near misses at either end with Hartson hitting the bar with a flashing header and Bobo Balde heading clear after Artur Boruc was stranded off his line.
With Hearts only drawing, this victory puts Celtic three points clear at the top and their ability to recover after going behind bodes well for their challenge.
Celtic slipped forward confidently from the kick-off and Maloney's pass to Hartson was just intercepted by Gary Caldwell at the expense of a corner.
Both teams started at a high tempo, but the home team had more of a cutting edge and Aiden McGeady was unlucky when he spun neatly on the edge of the box before hitting a shot straight at Zbigniew Malkowski in the Hibs goal.
The winger's next effort went too high, but already McGeady looked in the mood with his tricks and flicks troubling Hibs.
Boruc saved well from Garry O'Connor though the referee had already blown for handball by the forward.
Celtic were soon back on attack and Hartson did well at the back post to head back across goal, but Neil Lennon was denied by Steven Whittaker intervention.
Shunsuke Nakamura dispossessed Kevin Thomson just outside the box and from his pass Stilian Petrov chipped over the Hibs keeper, but Caldwell was able to scramble the ball clear.
Paul Telfer conceded a corner under pressure from Derek Riordan and then Boruc did well to hang on to a Thomson free-kick with Beuzelin closing in.
Hibs were finding it hard to create any clear cut chances despite good possession and were soon pinned back in defence.
Beuzelin was dispossessed by Petrov outside the area and the midfielder should have passed to the unmarked Hartson, but instead chose to shoot, his shot was deflected to Hartson but by that time the Hibs defence had recovered.
Another Celtic break ended with Hartson's pass to Maloney being cut out before Petrov was booked for a wild tackle on Whittaker.
Celtic opened the scoring five minutes before the break when Nakamura's free-kick, after McGeady was fouled, was bulleted home by Hartson with a trademark six-yard header.
Hibs were transformed in the second stanza having discovered a cutting edge that was missing from the first half. Petrov sliced a clearance behind and Thomson's corner was headed home by Beuzelin leaving the Celtic defenders arguing amongst themselves as to who should have picked him up.
Hartson tested Malkowski with a header before Celtic were stunned by going behind. Ivan Sproule, who had come on at the break for Riordan, was blocked out at the end of a mazy run, but Whittaker transferred the ball to the left flank where Fletcher was given enough time to angle his shot from 20 yards past Boruc.
Celtic looked stunned and Hibs nearly increased their lead shortly after. Thomson drifted past two defenders his pass found Fletcher inside the box, but the goalscorer delayed too long and the chance was lost.
The home team drew level when Maloney curled home a superb free-kick over the Hibs defensive wall. His kick was so well-taken and disguised that the keeper never moved.
O'Connor could have restored Hibs' lead when Beuzelin's pass released him between the Celtic rearguard, but the striker's shot lacked conviction.
Celtic scored the winner when Hartson, showing a striker's instinct, prodded home Maloney's shot which had come back off the post after he had flicked on Telfer's cross.
It was the end of the scoring though as Hartson was unlucky not to wrap up the points when his header came back off the bar from McGeady's free-kick.
Boruc did well to touch over Sproule's powerful shot, before the Polish keeper was left stranded when Ross Wallace was short with a pass back, but Balde had time to head clear O'Connor's shot over the keeper.
Man of the Match: John Hartson - The big Welshman came to the Bhoys' rescue with a double strike that earned a crucial three points.