The Shakers made the better chances against the League Two leaders but went down to a 53rd-minute goal by Darlington defender Alan White.
A mis-placed pass by White almost presented Andy Bishop with the opening goal for Bury, before Elliott Bennett sent a 20-yard strike fizzing over.
Darlington broke the deadlock shortly after the restart as the Shakers defence failed to deal with a free-kick into the danger zone and White swept the ball home.
Bury's best chance came when Andy Bishop hit the crossbar from a wide angle after latching onto Scott's long ball.
BURY: Brown 6; Scott 7, Futcher 7, Cresswell 7, Buchanan 7; Bennett 8, Baker 6, Barry-Murphy 7 (Haslam 61, 6), Jones 6 (Racchi 68, 6); Bishop 7, Morrell 7 (Hurst 90).
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