Ange Postecoglou's side cruised into the last four after beating Motherwell 4-0 at Fir Park on Wednesday evening, before Giovanni van Bronckhorst's men secured a slender 1-0 win against Championship outfit Dundee at Ibrox.
The Hoops, who are bidding to win their seventh Scottish League Cup in nine seasons, will now prepare for a semi-final clash at Hampden Park against 2011-12 winners Kilmarnock, who beat Dundee United 2-1 at Rugby Park on Tuesday courtesy of strikes from Kyle Lafferty and Danny Armstrong.
Rangers, who lost in last season's semi-final and who have not lifted the trophy since 2010-11, will take on six-time winners Aberdeen after they secured a 4-1 home victory against Partick Thistle on Wednesday, with Luis Lopes, Hayden Coulson and Ryan Duncan all on the scoresheet along with a Kevin Holt own goal.
Both semi-finals are scheduled to take place on the weekend of January 14/15 at Hampden Park, and the two victors will lock horns in the showpiece event on February 26.
Celtic made an electric start to their encounter with Motherwell and forward Kyogo Furuhashi could have scored a 30-minute hat-trick on another day, as he came close on three occasions to breaking the deadlock, including a speculative 25-yard strike which rattled the underside of the crossbar in the 19th minute.
However, Motherwell's backline was eventually breached just two minutes before half time when Greg Taylor played a neat one-two with Sead Haksabanovic before teeing up Liel Abada for a close-range tap-in.
Abada added a second 10 minutes after half time, placing a side-footed volley into the bottom corner after Josip Juranovic saw a long-range effort parried away by Motherwell goalkeeper Liam Kelly.
The Israeli winger then turned provider for Celtic's third goal on the hour mark, as he worked his way into the box and cut the ball back to Reo Hatate, who skillfully flicked the ball beyond Kelly from just a few yards out.
The Hoops capped off a comfortable victory in the 76th minute when Daizen Maeda capitalised on a mix-up in the Motherwell defence and set up Abada for his hat-trick, but his strike hit the inside of the post and fell to Furuhashi to tap into an empty net.
As for Rangers, Van Bronckhorst's much-changes side made a bright start to the contest with Dundee and opened the scoring in the ninth minute courtesy of a looping header from Steven Davis.
Scott Wright and Malik Tillman combined on the left flank before the former saw a powerful shot well saved by Adam Legzdins, and Davis was the first to react to guide his header into the back of the net.
The Gers dominated proceedings throughout, and although the visitors showed rare glimpses of a threat on the counter-attack, they failed to register a single shot on target.
Alfredo Morelos, Ridvan Yilmaz and Wright all came close to adding a second, but Rangers on the whole struggled to break down a resilient Dundee backline and they eventually did enough to edge through to the semi-finals.
All four semi-final teams will now turn their attention back to the Scottish Premiership on Saturday, with Celtic travelling to Hearts, Rangers welcoming Livingston to Ibrox, Kilmarnock playing hosts to Ross County, and Aberdeen visiting Motherwell.