The hosts missed several good chances while both sides had spot-kick claims turned down, but Sir Alex Ferguson's men held on for the three points in uninspiring fashion.
The Turkish champions had a penalty claim turned down inside the first minute and United soon made the most of that as Michael Carrick gave it the lead, but the visitors had their fair share of chances.
Nordin Amrabat and Hamit Altintop both struck the frame of the goal in the first half for Fatih Terim's side, while Nani missed a penalty in the second period, but the hosts were able to hold on for the three points.
With its noisy fans behind them, Galatasaray was straight on to the attack at the beginning and should have been given a penalty inside the first 40 seconds when Nemanja Vidic stretched to beat Burak Yilmaz to the ball after a poor back pass by Patrice Evra, but Wolfgang Stark bizarrely waved play on despite blatant contact from the Serbian.
It did not take long for United to capitalize on Stark’s error of judgement as it soon took the lead through Michael Carrick. The midfielder charged through the middle, and after an inter-change between Kagawa and van Persie, Carrick had possession again and tapped the ball in after cleverly taking on Fernando Muslera in the opposing goal.
The visitors were inches away from equalizing minutes later as United gave possession away sloppily in the center, allowing Nordin Amrabat to advance down the left before cutting inside and unleashing a curling effort, but the crossbar saved David De Gea from having to pick the ball out of his net.
United felt it should have had a penalty just after the half-hour mark when Kagawa was brought down in the area by Nounkeu, but replays showed Stark made the right decision, with the Cameroonian’s challenge perfectly timed.
With seven minutes left of the first-half Galatasaray hit the woodwork again as Altintop hammered a low effort against De Gea’s left-hand post following a short free-kick, and United held on to its 1-0 lead until halftime, if a little fortuitously.
The hosts started the second period in lively fashion and almost doubled their lead in the 47th minute as Nani and Kagawa combined excellently in the right side of the area, but Semih Kaya got a block in deny the Japan international.
A few moments later United had the perfect opportunity to score from the penalty spot after Rafael was brought down by Burak, but Nani had his spot-kick saved by Muslera and the Portugal international became the third player to miss a penalty for United already this season.
With 20 minutes to go there was a few second of madness, as first Evra had a shot put round the far post by Muslera, then an obvious foul on Nani in the penalty area was not given allowing the Turks to counter-attack, but De Gea pulled off an excellent double save to first deny Burak and then Emre Colak.
Substitute Javier Hernandez really should have sealed the win for United with five minutes to go as he got on the end of a lobbed pass from Nani, but the Mexican's left-footed strike went wide of the near post.
United had yet another penalty claim dismissed after Nounkeu brought down Antonio Valencia, but it ended up not mattering as the host crawled over the finish line.