Matt Smith powered home the decisive goal with just 11 minutes to go after he latched on to fellow substitute Kirk Millar's inswinging cross.
The victory - Johnson's second since taking over from Paul Dickov last month - keeps Oldham placed just outside the npower League One relegation zone.
Johnson said: "That was a totally heroic display from the lads.
"It was crucial we didn't concede towards the end and thankfully we managed to do that in what was always going to be a tough, tough game.
"It was difficult in the first half because Bury were giving it everything of course, but in the second period we restricted them to long-range efforts and in the end we've probably deserved our win."
For Bury boss Kevin Blackwell, it was a miserable day as he watched his gutsy side slide back into League Two in pouring rain at Gigg Lane.
It has been a traumatic week for the cash-strapped club although it was rumoured that potential new investors were sat in the stands.
Blackwell said: "Once again you've seen just how difficult it has been for us all this season. We've hardly got any players and we were only able to name four on the bench.
"Whatever happens from here on in though, these lads can go away and hold their heads up high. We've not always been able to call on the best quality, but in terms of effort they've never given this club less than 100 per cent."
- Sports & Recreation
- Paul Dickov