Dembele added the finishing touch to substitute Leigh Griffiths's cross in the 87th minute of the game.
Moussa Dembele grabbed a late winner as Celtic clinched a place in the Scottish League Cup final with a 1-0 defeat of Old Firm rivals Rangers at Glasgow's Hampden Park on Sunday.
Rangers looked a much improved side from the one hammered 5-1 at Parkhead in their last meeting with arch-rivals Celtic in August, but it was the Hoops who had the best chances, with Erik Sviatchenko's header ruled out for a foul before Scott Sinclair crashed a free-kick off the bar.
The game looked to be heading for extra-time before Dembele, who grabbed a hat-trick on his Old Firm debut, added the finishing touch to substitute Leigh Griffiths's cross in the 87th minute.
Celtic will join Aberdeen in the final after the Dons defeated Championship side Greenock Morton 2-0 in Saturday's first semi-final.
"I thought we were very dominant today in terms of chances and the quality of our game," Celtic manager Brendan Rodgers said.
"If the chances don't go in you are always vulnerable to the counter-attack but I thought that every element of our play was at a high level."
Rangers manager Mark Warburton was pleased with the improvement from his players.
"We're obviously feeling disappointed and frustrated but I think there are a lot of positives to take out the game.
"We've worked very hard to move forward as a team. The so-called gap is a lot narrower than people think and we have to learn from today's performance."
Rangers had selected full-backs Lee Hodson and James Tavernier down the right to deal with the pace of Sinclair and Kieran Tierney, which caused them so much damage in their September mauling by Celtic.
After a frantic start it was the Hoops who threatened first with Sinclair cutting in from the left to pick out Tom Rogic, whose deflected strike landed on the roof of the net.
Rangers were appealing for a penalty moments later when Barrie McKay went down under a challenge from Jozo Simunovic but referee Craig Thomson instead booked him for simulation, much to the fury of the Rangers players.
A slack pass was then intercepted by Dembele, who played Sinclair clean through, but Matt Gilks raced off his line to block his shot with his legs.
Gilks breathed a sigh of relief when his attempted pass went straight to Rogic in the box, but the Australian couldn't find the target with his effort.
The Rangers keeper then redeemed himself as he made a point-blank save to keep out a close-range header from Dembele.
The French striker was then lucky to avoid a booking for diving after he threw himself to the ground in the hope of a penalty as referee Thomson kept his cards in his pocket.
Celtic had the ball in the net three minutes after the break when Sviatchenko headed Sinclair's cross past Gilks but Thomson adjudged that the Dane had fouled Rangers defender Clint Hill in the build-up.
Sinclair and Rogic then combined to carve the Rangers defence open but Gilks blocked the English winger?s effort.
Rangers had yet to threaten Craig Gordon's goal but the best chance of the match then fell to Jason Holt when Tavernier found him in the middle but his goal-bound effort was well blocked by Simunovic with McKay sending the follow-up straight into the hands of the Hoops keeper.
Only the crossbar prevented Celtic from taking the lead in the 69th minute when Sinclair's curling free-kick rattled the woodwork.
With the game heading to extra-time Celtic made the breakthrough as Griffiths got behind Lee Wallace and his cross from the right was prodded home by Dembele under pressure from Rob Kiernan.