Derby had plenty of possession but never really looked like breaking through and Boro could have added further goals after the break.
The first half was desperately poor with the ball given away far too often, but the visitors went in front in the 38th minute with a moment of outstanding skill that was out of context with what had gone before.
Stewart Downing picked up a long ball down the left and delivered a superb cross on the run that Tuncay met with a left-foot volley that gave Stephen Bywater no chance.
The goal stung Derby into life and they almost equalised two minutes later when Kenny Miller turned on the edge of the area and sent in a shot that went inches wide of the right-hand post.
That was the closest Derby came to a goal and it was the visitors who almost scored again in front of their 2,000 fans early in the second half.
The impressive Tuncay broke away through the middle and played Downing in on Derby's right but his low drive was turned around a post by Bywater.
Middlesbrough went close again a minute later and should have made the game safe when Derby's defence was plunged into confusion by a cross that found George Boateng at the back post, but he headed against an upright.
Derby won a string of corners but the Boro defence coped well enough and Mark Schwarzer's handling was excellent on a bitterly cold afternoon.
The home side never gave up but even with Steve Howard on for his 500th League appearance, they lacked a cutting edge.
Towards the end, Giles Barnes fired over and also tested the keeper with a right-foot shot, but Boro deservedly held on for a victory that has given them renewed hope, while Derby are surely heading back to the Championship.