The Galatasaray forward and Stade Rennais man Mevlut Erding ensured that a Group D play-off spot will go down to the wire between the eastern European sides and Hungary
The visitors scored a goal in each half with Burak heading home before the break and Erding capping off a brilliant win in the late on, curling a left-footed shot into the net from inside the penalty area.
Pre-match favourites Romania failed to fulfil expectations throughout as interim coach Fatih Terim, in his third spell in charge of the national side, continued to inspire Turkey.
The vital win extends the nation's unbeaten run to four matches in qualifying and keeps them in the hunt for a play-off spot behind runaway Group D leaders Netherlands, who qualified thanks to their 2-0 win against Andorra and Romania's loss.
Tuesday's hosts do have the easier run-in compared to Turkey. Victor Piturca's men play Andorra and Estonia in their final two games and will still be hopeful of reaching the World Cup.
The first chance of the game came in the sixth minute when Mihai Doru Pintilii took advantage of some lax Turkish defending to sneak in behind, but his weak cross was deflected out by goalkeeper Volkan Demirel.
The visitors responded minutes later, though, through Burak, but the 28-year-old, under the close attention of the Romanian defence, side-footed his effort at the goalkeeper.
But Burak made up for his miss in the 22nd minute when he rose above two Romanian defenders to head home and give Turkey a surprise lead. The makeshift forward stood his ground to glance Caner Erkin's wonderful cross past Ciprian Anton Tatarusanu's outstretched palm.
Piturca made two substitutions after the interval, choosing to bring on Adrian Popa and Iulian Maxim as he tried to search for a way back into the match.
Turkey, though, could have easily been 2-0 up if Umut Bulut had been aware of how much time he had with only Tatarusanu to beat after he had been put through by Gokhan Tore.
Instead, the 30-year-old meekly hit his shot at Tatarusanu, even though he had team-mates in support.
Popa hit the post with a right-footed drive after 68 minutes as Romania pushed for an equaliser and Dorin Golan directed a late head straight at Demirel from a free kick.
Late substitute Erding then hit the second in injury-time when he evaded the raised legs of Romania's centre-halves to fire the ball home.