The victory boosted Galatasaray's hopes of reaching the knockout stage, putting them in second place in the group, behind United who had already qualified after four wins in their first four matches.
Burak put the Turkish champions ahead in the 53rd minute, heading into the top left corner above the head of Rafael after Felipe Melo had gone close seconds earlier and won a corner.
Hamit Altintop just failed to double Galatasaray's lead in the 78th minute with a powerful shot which flew off the crossbar.
"Beating Manchester United is important," said Galatasaray coach Fatih Terim. "We played brilliantly and unselfishly. My players played football in a way we have been expecting for a long time.
"We are very hopeful for second place," he added. "There is no guarantee in football but we will play well at Braga who no longer have any chance."
United manager Sir Alex Ferguson said: "Parts of the game were good but in the area which we were worried about - set pieces - we suffered.
"Other than that I was satisfied and it was a decent performance," Ferguson told the club website. "We had some good attacking play in the first half and at that point the only trouble they gave us was from outside the box. It was a very competitive, good game."
United's Darren Fletcher said the Istanbul pitch had made things difficult. "The biggest problem was that the pitch wasn't good at all," the midfielder told Sky television. "It was difficult to play one-twos."
