McLeish, who made four changes to the side which lost to Aberdeen in midweek, could only watch as his team fell to their third successive defeat in his final Old Firm match at Ibrox.
Maciej Zurawski's 12th minute strike turned out to be enough to separate the two sides and remove any doubt as to Celtic's total supremacy in the SPL, despite the fact that they were definitely firing on all cylinders in a match that will not live long in the memory.
Zurawski picked up a Stilian Petrov ball on the edge of the area and firing low into the right-hand corner of the goal past Ronald Waterreus.
Rangers were never overrun by the league leaders and could have equalised before the break but Thomas Buffel squandered his chance in front of goal.
The Light Blues did find themselves in danger again just before the break conceding a free-kick just outside the area after a heavy tackle on Shaun Maloney, but Dado Prso got back well to clear the ball off the line.
Rangers looked like a side more up for the game in the second half as McLeish urged his team forward.
Peter Lovenkrands joined the fray, replacing Nacho Novo, and sent a rising shot narrowly over the crossbar.
Sotirios Kyrgiakos sent a header inches past the post midway through the second half, but Artur Boruc seemed to know straightaway that it wasn't going to hit the target and never moved off his line
Celtic, despite only have a one-goal lead, never looked worried by the challenge. Rangers woes continued when Chris Burke picked up an ankle injury after all of the substitutes had been used.
Desperate Rangers pushed all of their resources forward in the dying minutes, but Boruc was equal to Marvin Andrews' shot.