Probably for the first time this season, Bury were under the cosh for long periods during the first half with summer signing Brown pulling off at least two top-class saves.
The Daggers, who were defending an unbeaten home record, were then stunned by two goals in five minutes from Stephen Dawson and Mike Jones.
Elliott Bennett sealed it late on after a comfortable second half for the Shakers, making Sam Saunders' stoppage-time strike merely a consolation.
It saw Bury bounce back from their first league defeat of the season last week and replace Daggers in second place.
"It was a great result because I think this is one of the hardest places to come to," said Knill.
"The first thing you have to do is stand up to the physical presence here and the three goals we scored were quality.
"Even though we were 2-0 up at half-time, I was pretty disappointed with our first half.
"Thankfully our goalkeeper was in top form. He made some unbelievable saves and kept us in it."
BURY: Brown 8; Scott 7, Sodje 6, Futcher 7, Buchanan 7; Bennett 7 Dawson 8, Barry-Murphy 7, Jones 8 (Racchi 83); Bishop 6 (Hurst 86), Morrell 7 (O'Grady 68, 7). Subs: Baker, Haslam.
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