A late third from Hamed Namouchi was perhaps harsh on an Austrian side that had contained Rangers well before fading away in the closing moments.
Not that that will cause McLeish much consternation as he examines the Group F standings to see his side sitting proudly atop of the five-man table.
Even before a ball had been kicked in anger, word filtering through that AZ Alkmaar had stormed to a 3-1 away victory over Amica Wronki served only to add to the importance of tonight’s game for both these sides.
Rangers, facing a potentially perilous trip to face the in-form Dutch team next week and looking to build on the 5-0 humbling of Wronki on their last Euro outing, started with just one enforced change to the side that defeated Celtic at the weekend, Bob Malcolm in for the suspended Alex Rae.
Grazer coach Walter Schachner had arrived in Glasgow with all the airs of the innocent abroad earnestly insisting that his side would be happy to return to Austria with a solitary point in the bag.
His starting line-up, however, seemed rather more adventurous than those modest expectations had suggested and included twin strikers Roland Kollmann and Mario Bazina spearheading an attacking 4-4-2 formation.
After the exhausting events of last weekend’s Old Firm collision, events here unfolded at a rather more sedate pace.
It was the visitors who enjoyed the best of the early chances with a string of shots from distance none of which, however, required Stefan Klos to get his gloves dirty.
For a team guarding a 14-match unbeaten run, Rangers’ start was surprisingly tentative with passes going astray and manoeuvres breaking down with alarming regularity.
It took 17 minutes for McLeish’s side to register as an attacking force when Malcolm’s thumping drive from all of 25 yards was unconvincingly parried by keeper Andreas Schranz.
For the second match in succession, a groin injury forced Gregory Vignal to hobble off earlier than intended and his replacement, Michael Ball, was soon involved in the action, crossing for Dado Prso to nod down for Marvin Andrews.
The giant Trinidadian did well to thump a fierce drive on target although on this occasion Schranz held the effort more effectively.
It would be the visitors, however, who would enjoy the best chance of a lacklustre first half when Bazina hared down the left wing before sending in a cross that found the unmarked Joachim Standfest just inside the Rangers’ penalty box.
Not for the first time a fantastic block from Klos denied the midfielder as Ibrox held its breath fearing the worst.
Kollmann knocked the rebound into the gaping net only for the strike to be ruled out for an earlier handball offence.
The decision by exotically-named referee, Olegario Manuel Bartolo Faustino Benquerenca, to play only four seconds of injury time said enough about his take on a dire opening period.
Rangers emerged reinvigorated after the break and within four minutes Fernando Ricksen picked out the darting run of Namouchi only for Gernot Plassnegger to haul him to the ground on the edge of the box.
The defender was duly booked although Rangers wasted the free kick, Malcolm ballooning an effort high into the stand.
The groans echoed once more around the stadium but the jeers would turn to cheers 13 minutes into the second half when the Light Blues eventually forged in front.
The accuracy of Ricksen’s set pieces has been the foundation for many of Rangers’ strikes in recent weeks and it was the same again on this occasion as the Dutchman swung in a tantalising free kick that arced narrowly over the head of Prso but found Novo unmarked at the back post.
From a matter of yards the Spaniard side-footed an effort low past the helpless Schranz.
Confidence restored, Prso then tried his luck from distance although the keeper was alert to smother his attempt.
At 1-0, Grazer’s ambition of taking a point was certainly still alive and Klos had to look alert to keep out a flashing header from top scorer Kollmann as the Austrians threatened once more.
Their menace was finally extinguished five minutes from time. Another pinpoint Ricksen centre was nodded back by Prso, Namouchi’s stabbed effort was parried only to Arveladze who scrambled a shot beyond Schranz.
The sheen on the scoreline arrived in the dying embers when Novo’s cross was turned in at the back post by Namouchi.
A sterner test will surely await in Alkmaar next week.
RANGERS (4-4-2): 1 Klos; 15 Khizanishvili, 4 Boumsong. 5 Andrews, 24 Vignal (Ball 29); 31 Namouchi, 12 Malcolm, 2 Ricksen, 26 Lovenkrands (Arveladze 77); 9 Prso, 10 Novo
Subs not used: Smith, Mladenovic, Hutton, Ross, Hughes
Booked: None
GRAZER AK (4-4-2): 1 Schranz; 26 Plassnegger, 4 Tokic, 5 Ehmann (Skoro 81), 18 Pogatetz; 7 Standfest (Dollinger 72), 23 Sick, 20 Aufhauser (Muratovic 69), 11 Amerhauser; 22 Bazina, 28 Kollmann
Subs not used: Almer, Potscher, Ramusch, Majstorovic
Booked: Plassnegger (49)
Man of the Match: Jean-Alain Boumsong