United skipper Jon Daly had given his side an early lead when he slotted the ball home after six minutes.
But Mohamed Chalali equalised with his first touch after 67 minutes only seconds after replacing Rory Fallon.
Iceland international Arnason gave the Dons their second away win of the season three minutes from time with a long-range strike which crashed off the crossbar and into the net to the delight of the large travelling support.
United kept with the same starting XI for the fourth successive match with Irishman Richie Ryan handed a place on the bench after gaining international clearance following his move from Sligo Rovers in November.
Aberdeen made two changes with Peter Pawlett returning following his two-match suspension for simulation and Fraser Fyvie started after recovering from a knee injury.
There was also a place on the Dons bench for goalkeeper Jamie Langfield following his return from a short spell on loan at Forfar as he continued his comeback after suffering a seizure in the summer.
The hosts almost broke the deadlock within three minutes of kick-off when Scott Robertson was afforded too much time to pick out Johnny Russell in the box but the United striker's header deflected off Andrew Considine and trickled past the post for a fruitless corner.
But Peter Houston's side did make the breakthrough with only six minutes gone when, after a poor punch out from Aberdeen goalkeeper Jason Brown, John Rankin's shot fell kindly to Daly and the Irishman slotted the ball home in off the post for his 14th goal of the season.
Aberdeen midfielder Robert Milsom, who was celebrating his 25th birthday, played an intelligent pass to Fallon on the edge of the area after 27 minutes but the New Zealander's low drive was ably handled by Dusan Pernis in the United goal.
The visitors almost handed United a second when Considine's back-pass nearly caught out Brown but the Welshman managed to clear the ball to safety before it crossed the line.
United had a strong appeal for a penalty turned down when Considine looked to have brought down Russell with the goal at his mercy just before the interval.
The first real chance of the second half fell to United after Gary Mackay-Steven was fouled by Pawlett on the edge of the area but Paul Dixon's shot from the resulting set-piece deflected wide.
Aberdeen made their first change after 60 minutes when Pawlett was replaced by defender Rory McArdle before Mackay-Steven made way for Willo Flood for United.
The Dons then introduced Chalali for Fallon and the change had an immediate impact when the Algerian headed the ball home with his first touch only seconds after coming on after McArdle had flicked on an Arnason long throw with 67 minutes gone.
Milsom should have given the Dons the lead after 74 minutes when, with the goal gaping, he somehow missed the target with a diving header from a perfect Scott Vernon cross.
United threw Ryan on for his first appearance in place of Stuart Armstrong with 10 minutes remaining as they looked to regain the advantage.
But it was the Dons who ended the game on top and Arnason scored a wonderful winner after 87 minutes with a thunderous shot from 30 yards, which Pernis could not keep out.