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 scoring a league goal.
Manager Chris Lucketti said: “I’m disappointed, again. We just didn’t create enough to win the game.
“Charlton have got a lot of pace in their side and we were wary of that because we’ve been conceding too many goals.
“With a flat back four, I think they would have had too much pace for us so I changed the formation to start with an extra defender.
“We knew that would mean we’d be pinned back at times but our turnover in possession needed to be better to get us up the pitch.
“James Hanson won most of his headers and we’d worked on getting Zeli Ismail and Danny Mayor in behind him but we didn’t do that enough.
“I think there is a mentality problem within the group, with the run that we’ve had and the lack of goals that we’ve scored.
“We just need a spark. The players are suffering in terms of confidence but a win will change that.”
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.