Only some sloppy finishing and industrious defending from the visitors prevented Rangers from adding to their tally and there were some nervy moments before Novo netted with just nine minutes left.
McLeish can, however, seek comfort in an otherwise accomplished performance rich in both effort and application.
After a sapping European tie on Thursday night, the Gers boss made four changes to the starting line-up in a bid to freshen things up.
One switch in particular was to pay dividends just 10 minutes into the match when Rangers went in front. Fernando Ricksen swung in a corner, Stephen Hughes drove in a low shot from the edge of the box and Andrews – in for Zurab Khizanishvili – thrust a leg out to divert the shot past Alan Combe.
It was the Trinidadian’s first goal since his summer move from Livingston and thoroughly deserved for all his honest endeavour and tireless effort alongside Jean-Alain Boumsong at the other end of the park. Peter Lovenkrands was another player relishing a rare start.
He had the ball in the net after just two minutes – the raising of the assistant’s flag prevented the Dane’s customary celebratory handstands – and was then denied what would have been an exquisite solo goal when he shot wide after a mazy dribble across the face of the penalty box.
In a half completely dominated by the home side, Novo and Steven Thompson both watched long-range efforts fly narrowly wide of target as Rangers tried, in vain, to finish their opponents off in the early stages.
Kilmarnock, for all their recent good form, offered little to show they had the nous or ability to overturn the early deficit.
The spotlight fell largely on Kris Boyd after his five-goal haul against Dundee United the previous week but his first real involvement of this affair was to pick up the first booking of the game for a clumsy challenge on Gregory Vignal.
Boyd enjoyed a competitive tussle with Boumsong throughout the 90 minutes, with the Frenchman in the main enjoying the upper hand. On the one occasion Boyd was able to free himself from his marker – latching on to a searching pass from Gary Locke – he squandered the opportunity, slicing his attempt well wide of Stefan Klos’ goal.
The teams emerged for the second half to be met with a sudden downpour but apart from the weather there was little change on the park as Rangers continued to bombard the Killie goal trying to squeeze out the second goal their dominance deserved.
After some indifferent early season form, Lovenkrands was proving to be a real menace on the left flank and, four minutes after the re-start, fired a shot just wide. Moments later and the Dane was inches away from connecting with Novo’s driven pass across the face of the Kilmarnock goal.
The Ayrshire side by this time were reeling on the ropes like a punch-drunk boxer as Rangers looked to administer what surely would have been a knock-out blow. Hughes was next to threaten the Killie goal with a volley that Combe did well to hold while Thompson could only watch on helplessly as he slipped a shot past the goalkeeper only to see it nudge the outside of the post.
At 1-0 Kilmarnock still had a chance of claiming something from this match and Klos did well to clutch a Danny Invincible drive after a crunching Locke tackle had dispossessed Hughes.
It was to be a rare moment of respite for the visitors and they were soon on the back foot again as Rangers poured forward once more. The bookies’ pre-match odds of 33/1 for first scorer suggested they didn’t see Andrews as much of a goal threat but only a stunning double block from Combe denied the robust centre-half his second goal of the game as he followed up a header with a stinging drive.
Moments later and he was threatening again with a header from a Fernando Ricksen corner that drifted narrowly wide All Rangers’ positive play, however, was almost undone after 76 minutes as Vignal nearly went from hero to villain as he dithered on the ball gifting Boyd possession.
The striker curled in a sweet shot that struck the inside of the post before being cleared to safety and moments later Rangers took full advantage of the let-off when they sealed the victory.
A long ball out of defence fell sweetly for Novo and he made no mistake with a powerful drive that gave Combe no chance.
RANGERS (4-4-2): 1 Klos; 21 Ross, 4 Boumsong, 5 Andrews, 24 Vignal; 2 Ricksen, 12 Malcolm, 27 Hughes, 26 Lovenkrands; 10 Novo (Arveladze 82), 19 Thompson (Prso 79) Subs not used: Smith, Kalenga, Davidson, McLean McCormack Booked: None
KILMARNOCK (4-4-2): 1 Combe; 2 Fowler, 4 Lilley, 6 Dindeleux, 22 Leven; 11 Invincible, 8 Locke, 7 McDonald, 12 Johnston; 10 Dargo (Nish 77), 9 Boyd Subs not used: Smith, Greer, Dillon, Murray, Dodds, Canning Booked: Boyd (18), Fowler (36), Locke (80)
Rangers’ Man of the Match: Marvin Andrews