كتب : Hatem Maher
Robbie Blake scored the winner after 19 minutes with a brilliant volley from inside the box.
United, who are bidding to become the first English club to win four consecutive league titles, wasted several clear-cut chances to level the score.
Midfielder Michael Carrick also had a penalty saved by Burnley keeper Brian Jensen just before the break.
Striker Michael Owen, who arrived in the close season, failed to impress and was replaced with Dimitar Berbatov in the second half.
"The chances we've had we should have won the match," United coach Alex Ferguson told Sky Sports.
"Burnley got the victory in 10 minutes in the first half. They played very well."
Liverpool atoned for their defeat to Tottenham in the league opener after crushing Stoke City 4-0.
Tottenham went top after recording an impressive 5-1 victory over Hull City, with Jermain Defoe grabbing a hat-trick.