Burnley took all three points midway through the first half as visiting goalkeeper John Filan could only parry a header from Ian Moore into the path of new Clarets skipper Branch, who found the roof of the net from close range.
Burnley's prized asset, striker Robbie Blake, was dropped to the bench by boss Steve Cotterill, pending a move to Premiership Birmingham City when the transfer window opens.
Wigan had previously made four separate bids to land Blake, but on each occasion they were rejected by the Clarets board.
WIgan boss Paul Jewell confirmed: "I made offers for the player, but Burnley are determined not to sell him to us.
"I know the player talked to some of our players, because he knows them and he wants to come, but they obviously don't want to let him go to us, so its not going to happen.
"We made what we thought was a fair offer, but Burnley do not want him to go to a first division club. I get the feeling if we offered £3million, they would not sell him to us."
Burnley boss Cotterill is now resigned to losing Blake, adding: "It's very disappointing for a manager, that's why I made him captain.
"I have got loads of time for him and it hurt a lot to put him on the bench. Any player that gets an opportunity to play in the Premiership, you can't hold him back.
"The only person who suffers is me, because I can't pick him any more.
"The two chairmen at Birmingham and our club are now talking about the future and I am sure things will materialise in the next few days."