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Bury were ahead less than 10 minutes in though as the visiting side forced a corner, which Rose got up highest to meet in the box from Chris Hussey, nodding into the top corner.
Rochdale came close to a quick equaliser when James Hooper controlled a long ball forward before racing away from the Bury defenders, leaving him with only the keeper to beat, but Daniel Bachmann parried the powerful finish clear.
Bury could have doubled the advantage midway through the first half when Donal McDermott beat two defenders to make his way into the Rochdale box, only to scuff his finish wide.
The ball was in the back of the net again shortly before half time as Hussey set up Peter Clark for a header, but a foul in the box saw the Bury midfielder's effort chalked off.
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The visitors continued to look the more likely at the start of the second half as Dodoo made a weaving run forward before cutting the ball back, but there was no-one there to make the most of his delivery.
Rose kept the pressure coming from Bury when he crossed the ball in for Clarke just after the hour mark, but Joel Castro Pereira managed to gather the ball at the second time of asking.
Another goal was disallowed for the visitors inside the final 10 minutes as Clarke burst forward from midfield to meet a long ball forward before coolly slotting past the keeper, but he was judged to be offside.
Rochdale desperately piled forward in stoppage time, attempting to force a replay, but the Shakers defended well enough to book their spot in the third round for the first time in eight years.
David Flitcroft's side will learn who they will face when the draw takes place tomorrow.