The Light Blues had not won at Parkhead for exactly three years and there seemed little chance of that jinx being broken in a disappointing afternoon for McLeish and his side.
Old Firm games are traditionally hectic affairs played more with passion rather than panache and this encounter was no different.
With both sides missing their influential strikers – injury prevented Henrik Larsson and Ronald de Boer from taking part – clear-cut chances were a rare commodity throughout a breath-taking 90 minutes.
Rangers, perhaps with one eye on playing for a draw, went with the pace of Peter Lovenkrands and Neil McCann up front alongside Shota Arveladze, with the Dane looking to add to the five goals he bagged against Celtic last season.
Service to the front trio was negligible, however, and it was the home side that enjoyed most of the possession as they clocked up a valuable win.
Barry Ferguson has often been a key figure in these clashes but he seemed to aggravate his pelvis injury in a collision with Sutton after 11 minutes and his influence was clearly muted after that.
The onus fell on Mikel Arteta to be Rangers’ main creative force and he tried desperately to bring his forwards into the game, often without success.
Sutton and Hartson were proving a real danger to the Light Blues’ defence and with the Celtic team and fans howling for every decision going it wasn’t long before the Parkhead side were benefiting from a string of set-pieces.
From one such corner, they should have gone in front after only 14 minutes when Stilian Petrov’s corner swerved right across the face of Klos’ goal to Hartson at the back post.
The Welshman slashed at the ball like a golfer trying to hack his way out of the rough and the ball trickled slowly past for a goal kick.
Jerome Bonnissel – in his Old Firm debut – was combining well with McCann down the left but Rangers seemed over-elaborate any time they looked in shooting range of Rab Douglas’ goal.
With Sutton falling over at every opportunity, both Fernando Ricksen and Moore were booked within five minutes of each other as Celtic continued to hem Rangers back.
Only a deflection from Amoruso took a Petrov shot wide of goal before Arteta had Rangers’ only decent chance of the half when he blazed a drive narrowly wide of Douglas’ goal.
The half ended with Celtic shouting for another penalty when Thompson went down with Moore but there looked little in it and Rangers were glad to reach the sanctuary of their dressing room with their goal still intact.
The second period started with the home side again in the ascendancy and Hartson missed another great opportunity when he headed wide from another Petrov corner.
With Celtic showing greater urgency and winning a boxful of corners, it was beginning to look grim for Rangers.
The match turning moment came just short of the hour mark as Arteta collected a ball in midfield and ran at the back-tracking Celtic defence.
An accurate pass would have sent Lovenkrands through on goal but the Spaniard hesitated and was dispossessed.
Thompson broke with the ball and crossed for Sutton who headed down to the unmarked Hartson who made it third time lucky by smashing the ball past Klos.
It was hardly a surprise giving the amount of Celtic possession but it was a real low blow nevertheless.
As time ticked away, Alex McLeish threw on Steven Thompson and Claudio Caniggia to try to rescue a point.
With Celtic defending deeper and deeper, there was little opportunity for either to make much of an impact.
The final chance came in the last minute, Amoruso’s blistering free kick palmed away superbly by Douglas.
The wait goes on.
Celtic: Douglas; Mjallby, Balde, Valgaeren; Agathe (Sylla 34), Lennon, Petrov (Macnamara 84), Lambert(c), Thompson; Hartson, Sutton
Subs: Broto, Crainey, Maloney
Booked: Sutton (45)
Rangers: Klos, Ross (Konterman 84), Moore, Amoruso, Bonnissel (Thompson 68); Ricksen, Arteta (Caniggia 74), Ferguson(c); McCann, Lovenkrands, Arveladze
Subs: McGregor, Malcolm
Booked: Ricksen (15), Moore (20), McCann (85)
Referee: Mike McCurry
Man of the Match: Jerome Bonnissel