Pat Fenlon's side knew they could go top of the SPL - if only for a few hours - with a win at Pittodrie but they got off to the worst possible start when Niall McGinn opened the scoring after Ben Williams had parried a Scott Vernon effort with only four minutes gone.
However a stunning equaliser from Irishman Eoin Doyle after 33 minutes ensured the sides went in level at the break in a half in which both sides struggled to cope with the swirling wind.
The winner arrived after 70 minutes when substitute Josh Magennis cut the ball back for midfielder Rae to net from close range.
Striker Leigh Griffiths returned to the Hibernian line-up after missing last weekend's 2-2 draw with Inverness - in place of Ross Caldwell.
Aberdeen made five changes from the side that defeated Dunfermline 1-0 in the Scottish Communities League Cup in midweek.
Skipper Russell Anderson, Ryan Jack, Stephen Hughes, Ryan Fraser and Vernon all returned with winger Jonny Hayes missing out through injury and Clark Robertson, Chris Clark, Cameron Smith and Magennis dropping to the bench.
Craig Brown's side went into the match on the back of an eight-game unbeaten run but their profligacy in front of goal had resulted in five of those matches ending in draws.
However the Dons found the back of the net in this encounter within four minutes when a hopeful ball upfield by Mark Reynolds landed somewhat fortuitously into the path of Vernon and, when his shot was only parried by Williams, McGinn reacted quickest to turn the loose ball home.
The visitors should have equalised after 11 minutes when Paul Cairney fed Griffiths but the striker's effort was well saved by Dons goalkeeper Jamie Langfield.
Hibernian full-back Alan Maybury was struggling to cope with the trickery of Dons winger Ryan Fraser and picked up the game's first booking for a crunching tackle on the youngster after 12 minutes.
The Dons could have doubled their advantage after 27 minutes when Ryan Jack picked out Vernon but the Englishman's goalbound effort was blocked by James McPake - with Aberdeen appeals that the Hibs skipper had used an arm to block the header falling on deaf ears.
A Stephen Hughes effort was blocked before Williams pulled off a fine save to deny Vernon, his former Colchester team-mate, as the Dons searched for a second.
The hosts' pressure continued to mount with Williams called upon to deny Stephen Hughes before tipping a Vernon shot over the bar.
But Hibs got back on level terms with a spectacular equaliser after 33 minutes when Doyle controlled a long ball upfield on his chest before producing a stunning strike from 25 yards which fizzed past Langfield.
The Aberdeen goalkeeper just managed to hold on to an awkward, dipping free-kick from Griffiths as an entertaining first half drew to a close.
Maybury, already on a booking, was withdrawn at half-time and replaced by on-loan Manchester City defender Ryan McGivern.
The Dons made a change of their own after 55 minutes when Fraser limped off and was replaced by Northern Irishman Magennis.
And within 60 seconds the substitute had a glorious opportunity to make an instant impact when McGinn swept a cross in from the left flank but the striker headed over from six yards out.
Cairney should have done better than blaze over from the edge of the area after being sent through on goal by Doyle before Magennis flashed a low, driven free-kick wide at the other end as both sides pressed for maximum points.
And it was the hosts who were rewarded with a second goal when Magennis used his pace to get by McPake before sending a tantalising ball across goal which was turned home by Rae.
Time ran out for Hibernian, who had a late penalty claim when Jack collided with Griffiths turned down, as their six-game unbeaten run in the SPL came to an end.