A Fernando Ricksen header early in the second half had drawn the sides level following Chris Sutton’s first half strike but a John Hartson effort was sufficient to sent Celtic through by the slimmest of margins.
Looking for his third successive Old Firm win, McLeish handed a debut to Bojan Djordjic, who lined up on the left of midfield, but kept Thomas Buffel waiting for his first Rangers start, the Belgian having to settle for a place on the bench.
Without a victory at this venue for almost five years, Rangers started comfortably – Hamed Namouchi saw a speculative effort blocked – before play descended into the sort of niggly midfield scrap that has become the norm whenever these two giants clash.
Even without the incendiary presence of the suspended Neil Lennon, referee Hugh Dallas had his work cut out right from the start, blowing his whistle more often than a sheepdog handler.
Hartson was proving to be Celtic’s main threat up front and, in their first threatening attack after 14 minutes, the Welshman found space to connect with Stilian Petrov’s searching cross only for his header to bang against the post, Marvin Andrews hooking the loose ball away.
Andrews’ next involvement was rather less productive however as he hurled himself headlong at Djordic’s corner, sending the ball spiralling over the crossbar and injuring both himself and Dado Prso in the process.
Rangers struggled on with nine men while the pair received treatment before Andrews returned to the fray rubbing his head gingerly and with a giant plaster stuck to the side of his face.
Seemingly impervious to pain, the giant Trinidadian was soon thundering into headers and tackles as if nothing untoward had ever happened.
After a positive start Rangers found themselves being gradually pushed deeper and deeper into defence as Celtic began to enjoy the lion’s share of possession.
Aiden McGeady wafted in a cross that Stefan Klos did well to palm away and when the rebound cannoned off Alex Rae the goalkeeper was there again to snare the ball away from Hartson’s toe.
If that was a lucky escape then the Light Blues were not so fortunate when Celtic went in front after 37 minutes
It was typical route one football from Martin O’Neill’s men; a Rab Douglas hoof glanced on inadvertently by Zurab Khizanishvili for Sutton to prod under Klos.
A wild Ricksen free kick that went hopelessly off target was Rangers’ sole response before the interval but the Dutchman would be more accurate 80 seconds after the re-start when the Ibrox side drew level.
It was the perfect way to drown out another torturous rendition of You’ll Never Walk Alone from the home support.
Rae fed Hutton wide on the right and when the young defender sent in an inviting centre, Ricksen stopped to nod a rare headed goal beyond the despairing leap of Douglas.
It was an equaliser warmly welcomed by the large travelling contingent but they had their hearts in their mouths just six minutes later when McGeady’s cross slipped through the hands of Klos and squirmed goalwards.
Khizanishvili reacted fastest to blooter the dropping ball away from danger and television replays confirmed the ball hadn’t crossed the line before the Georgian’s timely intervention.
Celtic continued to plug away and Sutton powered a header off the post from Alan Thompson’s corner kick before Hartson nodded over from right under Klos’ crossbar.
With Buffel on for Rangers and the game entering a lull phase, a replay was looking increasingly likely until Celtic went in front again after 77 minutes.
It seemed a harmless cross from Didier Agathe but a deflection carried it through to Hartson who rifled a low shot back across goal and into the bottom corner of Klos’ net.
It was a sucker blow for Rangers who had contained Celtic well and McLeish threw on Steven Thompson to try to once more rescue the tie.
Six corners in succession failed to yield a goal for the Ibrox side although Douglas had to be alert to pounce on an innovative back heel attempt from Nacho Novo.
Buffel had a late chance to salvage a replay but scooped his shot over the bar from Thompson’s knock down as Rangers’ Scottish Cup dreams died for another season at the home of their oldest rivals.
CELTIC (3-5-2): 20 Douglas; 6 Balde, 23 Varga, 16 Laursen; 17 Agathe, 19 Petrov, 4 McNamara, 8 Thompson, 46 McGeady (Camara 85); 9 Sutton, 10 Hartson
Subs not used: Marshall, Juninho, Lambert, Wallace
Booked: Hartson (90)
RANGERS (4-4-2): 1 Klos; 20 Hutton, 15 Khizanishvili, 5 Andrews, 24 Vignal; 31 Namouchi (Thompson 80), 2 Ricksen, 8 A Rae, 23 Djordjic (Buffel 67); 9 Prso, 10 Novo
Subs not used: McGregor, Malcolm, Ball
Booked: Namouchi (21), Ricksen (40)
Man of the Match: Fernando Ricksen