Tottenham Hotspur left it late to beat Birmingham City 2-1 to keep pace with Chelsea at the top. They both have a maximum 12 points with United on nine.
Nicolas Anelka, Michael Ballack and Ashley Cole scored either side of halftime for rampant Chelsea, who made it four wins out of four under new manager Carlo Ancelotti.
Later at Old Trafford, Andrei Arshavin gave Arsenal the lead but a Wayne Rooney penalty and Diaby's misplaced header turned the season's first heavyweight clash in United's favour.
Stoke City are well-placed in fourth after beating Sunderland 1-0 while Blackburn Rovers earned their first point of the season in a 0-0 draw against West Ham United.
Wolverhampton Wanderers and Hull City ended 1-1.
Chelsea's victory over Burnley made it nine consecutive league wins for the club after they ended last season on a five-game winning sequence under stand-in coach Guus Hiddink.
"It's important to maintain concentration, to maintain quick movement of the ball for 90 minutes. I am very happy every week because we are on the right road," Ancelotti told reporters.
"I am happy not just because we played well but because the (transfer) window closes in two days. We're doing very well with these players, I don't want to change, don't want other players."
Burnley, playing in the top flight for the first time since the 1970s, have already beaten Manchester United this season but were overwhelmed by Chelsea's midfield of Frank Lampard, Ballack, Deco and Michael Essien.
The breakthrough came seconds before halftime when Didier Drogba pulled the ball back for Anelka to score from close range. Chelsea made the points safe after the break, Ballack heading in Lampard's chip and Cole rifling in an unstoppable volley.
It is Chelsea's best start to a league season since 2005-06 when they won their first nine games and went on to capture the title under Jose Mourinho.
Arshavin's ferocious shot gave
