Mark Marshall broke the deadlock after 63 minutes and the Addicks were barely troubled as they made it back-to-back league wins for the first time since October.
Bury are now 12 points from safety after suffering their eighth straight defeat – and they have now gone 754 minutes without scoring a league goal.
Ben Reeves and Josh Magennis went close for Charlton before Bury’s James Hanson headed wide.
Keeper Ben Amos’ punt forward put Magennis clean through but his attempted lob was straight at Connor Ripley.
Reeves was then denied by Ripley before his inswinging free-kick deflected off defender Nathan Cameron and against the far post.
Bury captain Stephen Dawson and Charlton winger Stephy Mavididi both fired off target after the break.
But Joe Aribo turned Joe Skarz and cut back for winger Marshall to sidefoot home his first league goal for Charlton.
Mavididi then struck a post before Josh Laurent scuffed a shot wide as Bury pushed for a late equaliser.
Rangers manager Steven Gerrard is weighing up a move for Liverpool full-back Jon Flanagan, reports claim.
Scottish champions Celtic are reportedly ready to offer full-back Calvin Miller a new deal before sending him out on loan.
Everton target Jack Wilshere has sparked a scramble for his signature after confirming he is leaving Arsenal this summer.
Tottenham's hopes of landing Wilfried Zaha have been hit by the news Crystal Palace have offered him a bumper new deal to stay.
Ryan Giggs says Gareth Bale would be the perfect fit for Manchester United but has doubts over whether he will leave Real Madrid.
Manchester City's attempts to sign Riyad Mahrez could be in doubt again as reports claim Leicester only want a straight cash deal.