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.
But Bees boss Rosler insisted there was a lot more to come from his young side, who are simply suffering a slight blip in results.
His side should have been out of sight by the break, on-loan Fulham striker Trotta blazing a volley high and wide after a flowing five-man move.
Skipper Jonathan Douglas also headed straight at goalkeeper Scott Carson when Jake Bidwell's free-kick found him an acres of space at the far post.
"From a psychological aspect we are not the most experienced, but there is a lot more to come from my side," he said. "When we scored - and we have not won for a few games - the human instinct took over and we wanted to protect what we had.
"We are conceding too many goals, but I'm very relaxed as they are individual errors that we can resolve.
"Also and I'm not being disrespectful here, but a lot of the teams around us have been playing at their limit, but there is a lot more to come from us."