The Turkish side went into the game without a win in the competition so far and knowing only a win would realistically be enough to give them hope of going through behind group winners Manchester United.
And the visitors were immediately on the offensive with Emre Colak coming close in the sixth minute with a shot from 12 yards which flashed just wide of the Romanian goal.
Yilmaz opened his account in the 18th minute when Selcuk Inan played a lofted through ball and Yilmaz connected to head the opener into the top corner of the net.
Luis Alberto had the first real chance for Cluj in the 23rd minute when he brought a save out of Galatasaray keeper Fernando Muslera, then Modou Sougou shot wide as the home side threatened.
Cluj came close again at the start of the second half as Sasa Bjelanovic flashed a header just wide, before Sougou grabbed the equaliser in the 53rd minute after finishing off a pass from Rafael Bastos.
Bastos shot over as Cluj pressed for a second, but against the run of play it was Galatasaray who went back in front when Yilmaz netted his second from close range after a cross from Hamit Altintop was allowed to run all the way across the area by the home defence.
Mario Felguiras had to be alert to prevent Yilmaz netting his third in the 70th minute, but four minutes later it came as the striker slid the ball home from a tight angle to complete his hat-trick.
Bjelanovic came close again for Cluj in the dying seconds but Galatasaray coasted home for a victory which gives them renewed hope of muscling in on the knockout stages.