The home side were forced to play out the final 52 minutes with 10 men after Peter Vincenti's dismissal but were rarely troubled by an under-par Shakers.
Keith Hill's men shaded the first half and deservedly edged ahead in the 36th minute. Ian Henderson's corner was headed goalward by Vincenti but blocked on the line by 'keeper Trevor Carson, yet Hogan reacting quickest to poke the ball into the net from two yards.
Rochdale's elation was tempered two minutes later when Vincenti was dismissed for a second bookable offence. Having earlier left Nathan Cameron in a heap, his late challenge on Andy Procter saw Dale reduced to 10 men.
But Bury struggled to make their numerical advantage count after the break, with Dale stopper Josh Lillis barely tested. Substitute Temitope Obadeyi - a former Rochdale loanee - rolled a shot across goal but wide of a post while Jessy Reindorf sent a header on to the roof of the net.
Rochdale themselves still posed a threat going forward - William Edjenguele did well to turn Michael Rose's centre over his own crossbar from two yards out while George Donnelly saw a header saved by Carson.