A goal in each half by Derek Riordan (33, pen 80) cancelled out Lee Miller's equalising penalty (42).
Riordan opened the scoring after seizing on a slack backpass from Lee Mair before Miller got the Dons back on level terms from the spot after he had been fouled by Souleymane Bamba.
Riordan's winner came from a penalty, with Charlie Mulgrew sent off for handling on the line, although Steven Fletcher looked to have been in an offside position.
Aberdeen made two changes to the side that started against Celtic last Saturday, with Mulgrew and Sone Aluko replacing Stuart Duff and Derek Young as they looked for a first home win of the season.
Aluko looked dangerous whenever he got the ball and he used his pace and trickery to get past David van Zanten before delivering a dangerous ball Bamba did well to clear.
Despite their early dominance, the Dons almost gifted the visitors the lead after nine minutes when Zander Diamond misjudged his backpass and Jamie Langfield did well to spare the defender's blushes.
Hibs' first opportunity came in the 12th minute when Riordan's strike was deflected wide and the striker's resulting corner found Fletcher, who headed woefully wide.
Yves Ma-Kalambay came to Hibs' rescue minutes later when a glorious through-ball from Darren Mackie found the onrushing Richard Foster but the Hibs custodian used his 6ft 7in frame to prevent the opener.
Aberdeen were then cut open by a clever pass from Fletcher to Colin Nish, who shot wide when he really should have tested Langfield.
The relentless pace of the game continued with Miller and Mackie combining well before the latter produced a fine save from Ma-Kalambay.
However, it was the visitors who took the lead against the run of play thanks to some careless defending.
Riordan intercepted a short backpass by Mair before rounding the helpless Langfield and rolling the ball in from an acute angle.
Aberdeen almost replied instantly but Scott Severin's effort was saved by the impressive Ma-Kalambay.
Fletcher should have doubled the visitors' lead when he was presented with a glorious chance on his favoured left foot but he fired over.
Aberdeen got back on level terms soon after when Bamba brought Miller down in the area and the Dons striker sent Ma-Kalambay the wrong way from the spot.
The Dons seized the momentum and great play from Miller allowed Mark Kerr a chance on goal from the edge of box but his effort flashed wide of the left-hand post.
Nish was booked for dissent just before half time after referee Mike McCurry awarded Aberdeen a free-kick on the edge of the box when Miller was fouled by Bamba. Mulgrew's effort was well saved by the Hibs keeper.
Aberdeen started the second half brightest and Ian Murray failed to deal with Aluko's cross properly but Miller's strike was easily saved by Ma-Kalambay.
Jimmy Calderwood's half-time team-talk clearly had an impact as Aluko was next to have a shot at goal from 25 yards but he couldn't keep his effort down.
Steven Thicot almost put Hibs behind when his defensive header from a Mulgrew free-kick looped inches over his own crossbar.
Calderwood changed things up front for the Dons, with Chris Maguire coming on for Mackie after 66 minutes.
Ma-Kalambay again came to Hibs' rescue when he saved at the feet of Foster as the Dons continued to press.
Mixu Paatelainen made his first substitution of the day, with Dean Shiels coming on for Thicot, who looked to be suffering from cramp.
The switch seemed to galvanise Hibs, with Rankin crossing for Nish but his header went wide of the target.
Young replaced Jared Hodgkiss after 76 minutes as the Dons tried to regain the initiative.
However, it was Hibs who took the lead again from the spot as Langfield raced out of his goal to block a chance for Nish, leaving Mulgrew covering on the line.
When the defender handled Fletcher's flick on goal, McCurry had no option but produce a red card, although the Dons fans may have felt Fletcher was offside.
After netting his second, Riordan had a chance for the hat-trick with five minutes remaining but he could not find the target.