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.
United, knowing qualification was already assured, rested a number of senior players including Wayne Rooney, Rio Ferdinand, Ryan Giggs, Paul Scholes, David de Gea and Robin van Persie.
In their final match on December 5, Galatasaray, who are ahead of CFR Cluj on goal difference, will play Braga. The Portuguese side, who lost 3-1 to Cluj, are already out of the competition on three points.
After a cagey start, Galatasaray became more of a threat as the first half progressed, pinning the visitors back into their own half but creating few goalscoring opportunities.
Striker Burak headed over the bar from a Selcuk Inan free-kick in the 21st minute.
UNITED CHANCES
Galatasaray went close again some 10 minutes later when a strike from Hamit forced a save from Anders Lindegaard and United scrambled the ball away.
United worked hard to keep possession and neutralise any threat from Galatasaray and had chances of their own in the first half.
Eighteen-year-old Nick Powell, on his Champions League debut, saw his header fly off the crossbar in the 43rd minute.
Galatasaray started the second half in determined fashion and an Albert Riera volley from outside the penalty area was save by Lindegaard in the 49th minute.
United stepped up a gear and sought to find a way back into the game but Galatasaray survived a late bout of pressure to secure the points.
"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."
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