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.
England manager Gareth Southgate has claimed that the country needs to boot racism out of its own football before criticising World Cup hosts Russia.
Newcastle have been linked with a shock move for former Sunderland goalkeeper Simon Mignolet during the summer.