The visitors scored a goal in each half with Yilmaz 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 his Turkey team.
The vital win extends Turkey's unbeaten run to four matches in qualifying and keeps them in the hunt for a play-off spot behind runaway Group D leaders the Netherlands Holland, who qualified thanks to their 2-0 win against Andorra and Romania's loss.
Romania 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.
But Yilmaz 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.
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.
