The scoreline should have been more emphatic, with Christian Nade and Andrius Velicka both passing up several gilt-edged chances which could have put the visitors on easy street.
Dons boss, Jimmy Calderwood, recalled fit again Richard Foster in place of Steve Lovell for the a clash which could go a long way to determining Aberdeen's European aspirations for next season.
For Hearts, Deividas Cesnauskis failed a late test to be Steve Frail's only change from last week's derby win over Hibernian, his place going to on-loan Sheffield United striker Nade.
With only two minutes gone, Velicka sent Nade racing in on goal, but the striker took the wrong option and passed up an excellent chance to give the visitors the early advantage, when he fired weakly wide of the upright from eight yards.
Clever link-up play between Jamie Smith and Foster on 15 minutes saw the latter slip away from his marker, before firing in a fierce drive which was comfortably held by Hearts keeper Steve Banks.
Minutes later, Dons skipper Scott Severin had a free header from a Jeffrey de Visscher corner, but his effort from nine yards went harmlessly over.
Aberdeen had a major let-off again in 21 minutes when Nade turned Alexander Diamond on the by-line, before pulling a great ball back into the path of striker partner Velicka, whose 12-yard effort was cleared off the line by Jackie McNamara.
Hearts' physical presence was causing the Dons rearguard major problems. On 31 minutes a Saulius Mikoliunas free-kick wasn't dealt with properly, allowing Velicka to get in a downward header which Jamie Langfield got down to save at his near post.
The second half started with the Dons pushing for an opener. Chris Maguire's excellent crossfield ball dropped to de Visscher and he fed the ball back into the path of Barry Nicholson, whose crisp shot from 22 yards flew just over the crossbar.
Hearts then deservedly went ahead in 54 minutes. Debutant Stuart Duff's aimless pass in midfield bypassed the centre-backs allowing Nade a clear run in on goal.
The big striker atoned for his earlier miss to cleverly clip the ball over the advancing Langfield to open the scoring.
Aberdeen responded immediately when Lee Miller netted after clever wing play between de Visscher and Maguire, however, the referee correctly disallowed the striker's goal for offside.
The visitors were in total control and again should have added to their earlier goal on 78 minutes.
Ricardas Beniusis muscled Foster off the ball before playing a delightful ball into the area, where Andrius Ksanavicius' snapshot was only just blocked by the right foot of Langfield.
In the dying minutes a rasping shot from Foster saw Banks made a brilliant stop which was all Aberdeen had to show for a very poor second-half showing.