The result in west London meant that Shakhtar Donetsk finished the evening top of Group E following a 2-0 win at home to Danish competition debutants Nordsjaelland.
United took a seventh-minute lead against Galatasaray at Old Trafford when Michael Carrick swapped passes with first Robin van Persie and then Shinji Kagawa before skipping past goalkeeper Fernando Muslera and sliding the ball home.
The English giants - eliminated at the group stage last season - almost paid the price for taking their foot off the pedal, however, with Nordin Amrabat hitting the bar and Hamit Altintop clipping the post for the visitors before half-time.
Nani squandered an opportunity to give United breathing space when his 53rd-minute penalty was saved by Muslera, while the home side were indebted to a fine double save from David de Gea that saw him deny Burak Yilmaz and Emre Colak.
In the other Group H game, United's next opponents, Romanians CFR Cluj, won 2-0 at Portuguese side Braga courtesy of a first-half brace from former Braga forward Rafael Bastos.
Barca suffered an early blow at Camp Nou when Gerard Pique had to go off injured, but the Catalans - champions in 2006, 2009, and 2011 - went ahead when Cristian Tello curled a low shot home from the cusp of the 18-yard area.
A Dani Alves own goal gifted Spartak a 29th-minute equaliser, however, and the Russians sensationally took the lead just before the hour when Romulo beat Victor Valdes from Aiden McGeady's pass.
The Russians stood on the verge of a famous upset, only for Messi to equalise from Tello's lay-off in the 71st minute before heading in Alexis Sanchez's cross 10 minutes from time to hand Tito Vilanova victory in his first Champions League game as coach.
In the night's other Group G fixture, Celtic were held to a 0-0 draw at home to Benfica on the Scottish side's return to the group phase after a four-year absence.
Bayern Munich assumed control of Group F, winning 2-1 at home to Valencia - who lost on penalties to Bayern in the 2001 final - through goals from Bastian Schweinsteiger and Toni Kroos.
However, there was a disastrous start to the group phase for 2011 French champions Lille, beaten 3-1 at home to Belarusian outfit BATE Borisov, who had never previously won a game in the competition.
-AAP