Robinson could only name five substitutes but Addicks winger Mark Marshall claimed the game’s only goal after 63 minutes.
Charlton were barely troubled as they made it back-to-back league wins for the first time since October to climb back into the League One play-off places.
Boss Robinson said: “Today we could control our own performance, it was about us, so I’m more frustrated it wasn’t three or four.
“I would have loved to be having a cup of tea with 15 minutes to go.
“We need to take our chances, but we held on. We missed two or three one-on-ones but scored an amazing goal, so I’m really proud of them.
“We came here with 14 outfield players and only 12 of them were fit enough to play more than 15 minutes.
“Chris Solly and Jake Forster-Caskey told me on Thursday morning that they’d come up here even if they could only play five minutes.
“They’re not fit but they were willing to put their bodies on the line for the club.”
Ben Reeves and Josh Magennis went close for Charlton before Bury’s James Hanson headed wide.
Magennis then spurned a great chance before a Reeves free-kick was deflected against a Bury post.
Just after the hour mark, Joe Aribo turned Joe Skarz and cut back for Marshall to convert his first league goal for Charlton.
New loan signing Stephy Mavididi then struck a post before Josh Laurent scuffed a shot wide as Bury pushed for a late equaliser.
That made it eight straight defeats for the Shakers, who are now 12 points from safety and have gone 754 minutes without…