Tuncay capitalised on two bad defensive mistakes by Nicky Shorey and Steve Sidwell to give Boro their fine away success.
The Turkish striker pounced for the winner just two minutes from time to destroy Villa who had been looking to move into fourth place.
It was an outstanding Middlesbrough success as they produced an attacking performance which thoroughly deserved the three points.
Following Villa's midweek UEFA Cup success in Prague, match-winner John Carew was a late absentee with a hip injury and he was missed as Martin O'Neill's side struggled to break down the resolute visitors.
Gabby Agbonlahor was often playing as a lone striker although James Milner did move into the middle on occasions.
Neither side, however, could really dominate a game which consisted of some good individual play without either side producing the finishing touch in a quiet opening half-hour.
Boro demonstrated their potential with a good effort from Stewart Downing, while Villa's best opportunity was a free-kick from Ashley Young which was flicked over the bar by the in-form Ross Turnbull.
The central defenders on both teams were in commanding form with good tackles and interceptions. This cut down the goalmouth action until everything changed in the 34th minute.
A left-wing cross by Downing was misjudged by Shorey and Tuncay pounced to hammer a shot past Brad Friedel for only his second goal of the season.
Villa's response was immediate and within three minutes they were back on level terms.
Milner had drifted to the left and, after a Young shot had been blocked, the former Newcastle United player chipped a delicate cross to the far post where Sidwell was on hand to head home.
The pace of Agbonlahor nearly gave Villa a second goal. He easily went past Chris Riggott but Turnbull was able to narrow the angle as he blocked the close-range shot.
Turnbull again distinguished himself on the hour mark with a superb reflex save to keep out a first-time effort from Milner.
When play moved to the other end, Young, so often Villa's attacking hero, was their defensive saviour when he headed Riggott's powerful effort off the line with Friedel well beaten.
The miss of the game fell to Afonso Alves. Friedel failed to hold Didier Digard's shot but Alves, with an open goal in front of him, shot wide and was substituted seconds later.
However, Tuncay made no mistake with a low cross-shot after Sidwell had pushed the ball into his path after the former Reading and Chelsea player had attempted to clear.