Graham came close to putting the hosts ahead in the 11th minute when Bury failed to clear their lines, allowing the Blackburn forward to shoot, but the goalkeeper made a stop.
A few minutes later, the visitors had an opportunity thanks to some sloppy play from Blackburn, but George Miller's attempt was comfortably saved.
It took time for Rovers to produce some quality in the final third, but on the half-hour mark Elliott Bennett charged down the line before his shot dropped onto the roof of the net.
After the break, the hosts got their opening goal thanks to some link-up play with Bradley Dack and Graham, who headed home into the bottom corner of the net.
Harry Bunn starred for the visiting side, and he was almost rewarded with a goal when he picked the ball up inside his own half before storming down the field to unleash a curling effort, but his attempt went wide.
Bury were hit with a sucker-punch in the 67th minute, but it was their own doing after failing to clear a cross, which eventually resulted in Armstrong sliding the ball into the back of an open net.
The visitors remain second from bottom in the table, 10 points from safety.
BLACKBURN ROVERS (3-4-3): Raya; Lenihan, Mulgrew, Williams; Bennett, Smallwood, Evans (Payne 63'), Bell; Dack, Graham (Antonsson 63'), Armstrong (Nyambe 67')
BURY (4-4-2): Ripley; Edwards (O'Shea 78'), Cameron, Clarke, Leigh; Ismail (Hanson 68'), Laurent (Tutte 78'), Danns, Mayor; Miller, Bunn