Celtic's never-say-die spirit paid off as an injury-time header from Johan Mjallby won the home side all three points in an entertaining match.
When John Hartson put Celtic two up after only 23 minutes, it looked all over for the Edinburgh side, but Hibs also deserve credit for their fightback led by a double from Tom McManus.
Celtic made the perfect start when Hartson headed home Alan Thompson's cross in the first minute from Stephen Crainey's quickly taken free-kick.
Hibs thought they had equalised when McManus headed home a hopeful punt, but he was adjudged offside.
Shaun Maloney was relishing his chance to start a match and twice he went close with snap shots. The small striker then turned creator with a corner which Hartson glanced into the top corner from six yards to put the home side two ahead.
Joos Valgaeren showed up the nerves in the Celtic defence as he waited for Javier Sanchez Broto to come for a loose ball. This uncertainty was punished minutes later when Broto waited too long to clear Didier Agathe's backpass allowing McManus to charge down his clearance before rolling the ball home into the empty net.
Celtic stormed back and from Thompson's cross, Hartson's netbound header was brilliantly parried by Daniel Andersson, before Ian Murray completed the clearance via his own crossbar.
It was the Edinburgh side who took the match to Celtic at the start of the second half and Broto partially redeemed himself with fine saves from Murray and Derek Townsley before Hibs gained a deserved equaliser with a superb goal.
McManus stepped inside Valgaeren before curling a tremendous drive into the top corner of the net from fully 25 yards.
Celtic boss Martin O'Neill sent on Momo Sylla for the final ten minutes and Hartson's volley was well blocked by Andersson.
The Hibs keeper then brilliantly blocked a Mjallby headed, but just when it looked like Hibs would hold out for a share of the points, Mjallby powered home Thompson's corner at the back post.