An unfortunate own goal by Hoops defender Daniel Majstorovic after just four minutes gave the home side the interval lead.
However, a fine James Forrest double and goals from Anthony Stokes and Gary Hooper turned things around in the second half to put the Parkhead men into the hat alongside Kilmarnock and First Division sides Ayr and Falkirk.
To compound a miserable night for Hibs their wide man Ivan Sproule, whose shot had cannoned off Majstorovic for the goal, was sent off by Willie Collum in the 71st minute after picking up the second of two yellow cards for a foul on Mark Wilson.
In the end it was an easy night's work for Neil Lennon's side but he will be desperate to eradicate his side's growing habit of giving the opposition the lead.
With injuries to Glenn Loovens and Charlie Mulgrew, the Celtic boss was forced to reshuffle his central defence yet again.
Thomas Rogne made his first appearance of the season and Majstorovic started with Wilson taking over from Victor Wanyama, who was named as substitute.
Goalkeeper Mark Brown, David Stephens, Victor Palsson and Ivan Sproule came in for Hibs with Graham Stack and Danny Galbraith on the bench and Sean O'Hanlon and on-loan defender Richie Towell - not allowed to play against his parent club - missing altogether.
Wilson was Celtic's captain for the night in the absence of Mulgrew but eyes were on Rogne and Majstorovic.
The latter, unfortunately for him, was involved when the home side took the lead in the fourth minute.
Leigh Griffiths' corner from the right was blocked at the front post by Adam Matthews but the ball went only as far as Sproule who quickly drove it across goal - and his shot came off the hapless Sweden stopper and into the net at the far post.
Hooper slid Beram Kayal's cross from the right wide from close range in the 20th minute and five minutes later former Hibs striker Stokes tried his luck with a free-kick from distance but although he generated good power, the ball flew just over the bar.
Two minutes later, as visiting fans continued to arrive, the Irishman struck the outside of the post from the edge of the box with a left-footed drive after being set up by Kayal, then headed a Ki Sung-yueng corner over the bar.
Five minutes before the break Celtic keeper Forster made a good save when he came out to block Griffiths' shot after the striker had been sent clean through on goal by O'Connor with the Hoops defence missing.
Less than two minutes after the restart Forrest levelled. The Scotland international took a pass from Hooper on the edge of the box and skilfully turned defender Paul Hanlon before sending a wonderful drive past Brown and into the far corner.
The momentum had shifted back in favour of the visitors and in the 58th minute Forrest drove towards the Hibs box from the right. He evaded some weak attempts to tackle and worked a one-two with Stokes before driving low past Brown to make it 2-1 to the visitors.
Hibs looked to have lost their belief and in the 64th minute the tie appeared to be settled when Stokes rose just six yards out to head in a Ki corner from the right.
Moments later, with the away fans still celebrating, Stokes' turn and drive had Brown diving low to his right to prevent the ball speeding in at the far post.
In the 69th minute, with the home defence AWOL, Matthews sent Hooper racing through on goal unhindered and he calmly guided the ball low past Brown to seal the win.
Two minutes later, Sproule, booked earlier after a tussle with Kayal, was shown a deserved second yellow card after clattering Wilson outside the box - and walked off without much protest.
It was a case of damage limitation for the bedraggled Leith side who, in the end, were glad the margin of defeat was not greater.