Rangers created the best opportunity of the opening ten minutes when a weak punch clearance by Ryan Esson fell to Fernando Ricksen, but his hurried effort was easily knocked off the line by Scott Severin.
Minutes later Dado Prso jinked inside Zander Diamond before picking out strike partner Nacho Novo on the edge of the area, his first-time effort only just cleared the crossbar with Esson struggling.
Aberdeen took an unlikely lead on 30 minutes when a Gary Dempsey free-kick caused panic in the Rangers penalty box. Ronald Wattereus came, but failed to collect, the ball breaking to Russell Anderson who gleefully headed home from six yards.
With their tails up, Aberdeen extended their lead on 37 minutes, Jamie Smith and Barry Nicholson tore at the Rangers rearguard before the latter picked out Steve Lovell, who stepped inside before rifling home from 14 yards.
Rangers' response was immediate as they forced a corner at the other end. Ricksen's flighted ball into the area found Prso who had all the time to nod home past an unprotected Esson from six yards.
Rangers drew level on 49 minutes when second-half substitute Steve Thompson clipped in a left-foot cross at the second attempt which fell to Peter Lovenkrands, whose acrobatic overhead kick flew past a stranded Esson to bring his side level.
Ricksen needlessly picked up his second yellow card when he went in recklessly on Kevin McNaughton, leaving referee John Underhill no alternative but to dismiss the Dutchman.
Aberdeen thought they had regained the lead on 53 minutes, Dempsey's deep cross caused the Rangers rearguard all sorts of problems, Chris Clark hooking the ball back into the area for Severin to knock the ball into the path of Lovell, who stabbed the ball home only for the goal to chalked off for offside.
Just as the game seemed to be heading for a share of the spoils, Aberdeen came up with a timely winner.
With 88 minutes gone, Smith picked up a loose ball 25 yards out with the Rangers defence backing off he advanced a couple of yards to fire an unstoppable shot into the corner of the net.