Robbie Blake struck a superb volley from outside the box in the 19th minute and United wasted the chance to equalise when Michael Carrick saw his penalty well saved by Brian Jensen just before the break.
United piled on the pressure in the second half but an inspired Jensen and some wasteful finishing kept them at bay.
Liverpool put an opening defeat to Tottenham Hotspur behind them with a 4-0 victory over Stoke City with Fernando Torres opening the scoring.
Tottenham are the new league leaders after thrashing Hull City 5-1 with England striker Jermain Defoe grabbing a hat-trick.
They lead Chelsea on goal difference after Carlo Ancelotti's side made it two wins from two with victory over Sunderland on Tuesday. Arsenal are third on three points having played only one game.
Burnley, who won the second of their two league championships in 1960, are back in the top flight for the first time since 1976.
"The chances we've had we should have won the match," United manager Alex Ferguson told Sky Sports news. "Burnley got the victory in 10 minutes in the first half. They played very well."
Burnley took their surprise lead with Blake's beautifully taken volley and escaped an equaliser minutes before halftime when Blake tripped Patrice Evra in the box only for Carrick to miss the spot kick.
Michael Owen, substituted in front of England coach Fabio Capello on the hour after failing to make an impact, made way for Berbatov who improved United's attack but Burnley held on.
Liverpool went ahead at home to Stoke early on when Spanish striker Fernando Torres met Steven Gerrard's cross to score before Glen Johnson grabbed a second just before the break.
Great work from captain Gerrard allowed Dutchman Dirk Kuyt to add a third on 78 minutes and David Ngog added a fourth late in the game.
Birmingham City's 1-0 victory over Portsmouth, with a last-gasp penalty from James McFadden, meant there is yet to be a drawn match in the new Premier League season.
(Editing by Kevin Fylan)