Yilmaz climbed highest to meet a 54th-minute corner to inflict upon United their first defeat in Group H.
With United having already qualified as group winners, the win leaves Galatasaray hanging on to second spot before the final matchday in a fortnight.
With top spot already secure, Sir Alex Ferguson left many of his stars at home, with Nick Powell handed his first European start and a welcome return for Phil Jones after a knee injury.
The hosts had the better of the early exchanges, with Anders Lindegaard keeping out Yilmaz's drive from distance, before the Turkish forward headed over when unmarked midway through the first half.
United grew into the game and came closest to opening the scoring when Powell headed Tom Cleverley's corner against the bar two minutes before half-time.
Albert Riera tested Lindegaard again four minutes after the interval with a volley from distance which the United keeper did well to hold.
Lindegaard made an even better save shortly after to turn Felipe Melo's header over the bar, but from the resultant corner in the 54th minute, Galatasaray took the lead.
Selcuk Inan's delivered an outswinger from the right which Yilmaz attacked, outmuscling Darren Fletcher on his way to planting a header into the top corner, over Rafael on the far post.
United continued to be a threat on the counter attack, but Fernando Muslera remained relatively untroubled in the hosts' goal.
Ferguson threw on Ashley Young and Federico Macheda for the last 15 minutes, but Galatasaray almost doubled their lead moments later when Hamit Altintop's 25-yard drive was brilliantly tipped onto the bar by the diving Lindegaard.
United spent much of the last ten minutes on the front foot, with Josh King replacing Danny Welbeck, but Galatasaray held on and now travel to Braga on the final matchday in possession of the second qualifying spot in Group H.