The Scottish champions had returned to Glasgow chastened and weary following their 3-0 Champions League defeat by Bayern Munich in Germany on Wednesday but a first-half double by right-back Mikael Lustig looked to have them on easy street.
However, former Celt Anthony Stokes reduced the deficit and changed the complexion of the match just before the hour mark when he scored from the spot.
As Hibs went all out for the leveller, Celtic substitute Dembele restored their two-goal lead and after 19-year-old substitute Oliver Shaw made it 3-2, the French striker grabbed a breakaway fourth to set up a final meeting with Motherwell or Rangers, who play their semi-final on Sunday.
An "absolutely delighted" Rodgers said: "It was a brilliant result for us, especially after the exertions of midweek and against a Hibs team which, as we've seen a few weeks back (in a 2-2 league draw), can be a real threat.
"It was clear that there was a little bit of fatigue in our play at times which is only natural.
"That affects the technique but I thought that mentality and concentration of the players (came through) and to end up getting four goals is absolutely brilliant.
"So I tip my hat to them because it could have been a real banana skin for us as I am sure a lot of people might have thought."
Rodgers, however, was less impressed with referee Kevin Clancy, who, after consulting assistant Frank Connor, deemed defender Dedryck Boyata's tackle on Martin Boyle to be a penalty.
The Northern Irishman said: "My initial reaction at the time was that Dedryck Boyata recovered well.
"Their boy is running frontal and he comes in and gets his toe to it and the ball is misdirected.
"So there are two poor decisions.
"One by the referee, who was on his way back and looking for help from his linesman who is obviously facing it, and how he gave the penalty I will never know.
"It gave them a bit of oxygen and made us focused and I thought the players pulled themselves back really well."
Hibs boss Neil Lennon was happy with the way his side responded after giving themselves "a mountain to climb".
The former Celtic player and manager said: "It was two poor goals in the first half and then we were playing catch up.
"We weren't getting bombarded. They were dominating possession but that was fine, we were happy to let them have it in certain areas.
"We had the opportunity to clear the first goal, we didn't and my goalkeeper should do better at the second.
"But we came back very strongly in the second half and almost got something out of the game.
"But you can't give a head start to a team of that calibre.
"Our football was good and maybe those little bits of quality they have over us was the difference."
