The Shakers managed just two efforts on goal at Griffin Park, but scored with both as the Bees had to settle for a point.
Tom Soares and Nicky Ajose were the heroes for the visitors, whose mini-revival began with a win over promotion-chasing Doncaster last week.
Goals from Harry Forrester and Marcello Trotta twice gave the Bees the lead, but Uwe Rosler's Cup heroes looked edgy throughout and sections of the home crowd booed them off after they were repeatedly thwarted by Bury's well-drilled rearguard action.
Blackwell was delighted with his side's determination to "dig in and make sure they got something out of the game".
He said: "We had two clear-cut chances and took them both and you can't argue with that. We were clinical and bided our time.
"We knew it would be a very tough place to come and get anything and we knew this month, with Doncaster, Sheffield United and Coventry, would be a big test.
"When I got to Bury we had no chance of staying up and now we are one win away from being clear of the relegation zone.
"That's a remarkable achievement and it's all down to the players. We knew we had to show Brentford respect and in the second half our patience paid off."
- Sports & Recreation
- Marcello Trotta
- Tom Soares
- Uwe Rosler
- Nicky Ajose