Rangers lost more ground on the pace-setters at the top of the SPL after Aberdeen claimed their first point at Ibrox since September 1997.
Steven Thompson came close to opening the scoring in the first half but his shot was touched wide by Ryan Esson.
Esson again came to the rescue after the interval, when he blocked a point-blank effort from Dado Prso.
The result leaves the defending Champions in fourth place, 12 points adrift of leaders Hearts.
Stefan Klos was back between the sticks for the Champions, as Ronald Waterreus dropped to the bench for the visit of Aberdeen to Ibrox.
But the German had little to do on another frustrating afternoon for the defending champions.
Rangers threatened when Barry Ferguson set up Fernando Ricksen but the ball was cleared by the Aberdeen defence, and fell to the feet of Thompson only for his shot to be touched wide by Ryan Esson.
At the other end, Gary Dempsey tried to release Jamie Smith but Klos raced off his line to smother, prompting rapturous applause from the home support.
A smartly-worked one-two between Ferguson and Thompson allowed the Rangers captain a chance at goal but his long, low effort was just inches wide of the target.
The home side quickly set about putting Aberdeen under pressure within seconds of the restart, when Ferguson's short pass set up Dado Prso but the Dons goalkeeper did well to block the point-blank shot.
With the Ibrox side becoming restless at the lack of goals, Olivier Bernard tried his luck with an opportunistic drive from the edge of the area but the shot never really troubled Esson.
Rangers continued to pound the Aberdeen goal and next up to have a go was Ian Murray but both efforts cannoned off the resolute Dons defence.
Francis Jeffers - still to open his goalscoring account for Rangers - was thrown into the action for the last 20 minutes, taking the place of Alan Hutton.
But it was Prso who should have broken the deadlock.
Burke's inswinging free-kick found Sotirios Kyrgiakos at the near post and he played the ball to the feet of the Croatian hitman two yards out, but Esson managed to block when it looked certain Prso would find the back of the net.
In the end, there was no win to celebrate for birthday boy Prso, following a game which will pile the pressure on manager McLeish.
Rangers: Klos, Hutton (Jeffers 70), Rodriguez, Kyrgiakos, Bernard, Ricksen, Ferguson, Hemdani (Murray 27), Lovenkrands (Burke 45), Prso, Thompson. Subs Not Used: Waterreus, Andrews, Namouchi, McCormack.
Aberdeen: Esson, Clark, Diamond, Anderson, Byrne, Nicholson, Dempsey (Foster 81), Severin, Muirhead, Smith (Mackie 69), Crawford. Subs Not Used: Langfield, Winter, Stewart, Considine, Baas.
Ref: S Dougal