The victory, coupled with Dundee United's home defeat to Rangers, was enough to secure fourth place for Jimmy Calderwood's men.
The Dons took the lead in bizarre circumstances when the Hibs goalkeeper Yves Ma-Kalambay kicked the ball off Lee Miller and could only watch in agony as the ball rebounded off the striker's back and into the net.
The home side doubled their advantage when Charlie Mulgrew, only playing after successfully appealing the red card he received at Rangers last week, curled an exquisite free-kick into the net after 45 minutes.
Hibernian gave themselves a lifeline just before the half-time whistle when Derek Riordan cut inside and fired beyond Jamie Langfield.
The Dons held on for the victory in a tense second half, which saw the visitors reduced to 10 men when Ian Murray was sent off for a second booking and Sone Aluko miss a late penalty.
Stuart Duff started in place of Mark Kerr, who missed out as a result of injuries sustained in an alleged incident outside a nightclub earlier this week, while Hibernian named the same eleven who drew 0-0 with Celtic last Sunday.
The Dons had not beaten the Easter Road side this season but made a lively start when Peter Pawlett's left-footed volley was blocked before skipper Scott Severin's snap-shot deflected wide of the target.
Then came the fortunate opener, Ma-Kalambay kicking the ball off Miller, who was standing only a couple of yards away when the ball rolled into the net off his back.
The away side complained to referee David Somers that the striker had been impeding the keeper but the goal was allowed to stand.
Severin, in what may turn out to be his last appearance for the Dons, stung the palms of Ma-Kalambay with a rasping drive from 30 yards after 24 minutes.
The Aberdeen captain looked determined to get his name on the scoresheet but was thwarted again three minutes later when the Hibernian keeper clawed his downward header to safety.
The Belgian stopper then pulled off a fantastic diving save to prevent Gary McDonald adding a second and from the resulting corner Pawlett's deflected effort rebounded off the post as the home side enjoyed a period of dominance.
And the Hibs keeper was picking the ball out of his net again before the break when Mulgrew curled an exquisite free-kick over the Hibs wall and into the back of the net from 25 yards.
However Riordan made it interesting when he cut inside and fired the ball low into the back of the net from just inside the box.
Aberdeen came out for the second half knowing that results were going their way elsewhere and a win would guarantee them a European berth.
Langfield, who was this week voted Aberdeen's player of the season by the club's supporters, had to rush off his line to gather the ball at the feet of Steven Fletcher after an enticing through ball from Riordan.
Miller had a chance to grab his second off the afternoon from a free kick on the edge of the box after 63 minutes but his powerful drive cannoned off Rob Jones and away to safety.
The striker spurned a great chance moments later when Duff centred the ball into his path but Aberdeen's top scorer shot straight at Ma-Kalambay.
Hibernian were reduced to ten men after 69 minutes when Ian Murray, who had already been booked, was shown a second yellow for a challenge on Pawlett.
Mulgrew sent in another teasing delivery after 73 minutes but Miller could not apply the finishing touch.
With 10 minutes to go Aberdeen freshened things up by bringing on Michael Paton for the tiring Pawlett, while Jonatan Johansson replaced Fletcher for the visitors.
Aluko should have put the game beyond doubt but his penalty was saved by Ma-Kalambay with less than four minutes to go following a handball by Johansson but the Dons held on for all three points.