A goal from Bert Konterman after 15 minutes of extra time put Rangers in the final, where they will meet the winners of tonight's game between Hibernian and Ayr United.
It was a typical Old Firm clash with thrills aplenty. Rangers took the lead on the stroke of half-time, but then missed a penalty in the second half.
But Konterman grabbed the winner. An Arveladze move was blocked by Mjallby, but the ball fell to Konterman and he rattled it into the back of the net from 25 yards.
Before the clock had reached 10 minutes chances had come and gone with astonishing regularity. Celtic, after just 60 seconds, came close to taking advantage of a slack Arthur Numan pass-back and it required a timely block from Lorenzo Amoruso to stop John Hartson.
The tie kept bubbling along in a frantic manner and Shota Arveladze's touch from a Ronald de Boer flick looked as if it might count until Bobo Balde cleared. Then Henrik Larsson, of all people, missed an outstanding chance after seven minutes when Alan Thompson's cross drifted over Amoruso's head to give the Swede an opportunity, but the slightest of touches from Stefan Klos diverted his shot past.
After all that there had to be some respite and as the game settled to some kind of pattern, Celtic looked a touch sharper and more dangerous.
It was all Rangers could do to put together an attack and they relied mainly on the touch of De Boer and pace of Peter Lovenkrands, although Fernando Ricksen moved forward on the half hour and wasn't too far away with his shot.
That encouraged them to go forward more often and they opened the scoring after 45 minutes, ironically after seeming to waste a free-kick following a Balde foul on De Boer. Celtic failed to clear the danger, Barry Ferguson headed forward, Arveladze got a touch and Lovenkrands stole behind Johan Mjallby to scramble the ball home.
It took referee Kenny Clark all of 51 minutes to bring out his book, but when he did he made up for lost time. First, Ferguson for complaining about a Neil Lennon tackle on Ricksen and then the Irishman for voicing his opinion. They were followed four minutes later by Arveladze for a wild lunge on Agathe.
There was a major blow for Rangers after 62 minutes when Craig Moore, Klos and Larsson collided and the Australian was stretched off. Then Konterman was booked for a bad tackle on Agathe as tempers became frayed.
Celtic were on level terms after 74 minutes. Alan Thompson sent in a corner that Rangers failed to clear and after Mjallby tried and couldn't score it was left to Balde to get the final touch.
The Celtic defender went from hero to villain when he brought down Lovenkrands in the penalty area. But he was a relieved man when Arveladze's penalty struck the top of the bar and went over.
Despite appearing to have a large hole in his leg, Amoruso had the injury treated and continued bravely in the heart of the Rangers defence as extra-time began.
A Lubo Moravcik free-kick six minutes into the extended period gave Celtic the first chance and the kick was nodded on by Valgaeren to Agathe in the box but the Rangers goalkeeper managed to block it.
Balde found himself lucky not to be red-carded for a hefty challenge on Flo after eight minutes. But the Norwegian recovered quickly and tried to set up Ferguson a minute later but the skipper's shot deflected wide off Balde.
After 11 minutes Arveladze cut the ball back to Numan just inside the area but the Dutchman fired into the side-netting.
RANGERS: Klos, Moore (Vidmar 63), Konterman, Amoruso, Ricksen, de Boer (Flo 52), Ferguson, Numan, Caniggia, Arveladze (Hughes 118), Lovenkrands. Subs Not Used: McGregor, McCann. Booked: Ferguson, Arveladze, Konterman, Klos, Caniggia, Numan. Goals: Lovenkrands 45, Konterman 105.
CELTIC: Douglas, Mjallby, Balde, Valgaeren, Agathe, Lambert (Moravcik 72), Lennon, Petrov (Petta 105), Thompson, Hartson (Sutton 65), Larsson. Subs Not Used: Gould, Crainey. Booked: Lennon, Balde, Larsson, Sutton. Goal: Balde 74.
Referee: K Clarke (Scotland).
