A new-look Cherries side led 2-0 at the interval after Sammy Igoe and debutant Joe Partington struck late in the first half.
But for all their efforts after the break, the Shakers were unable to avoid just their second away defeat of the season, although they remain second in League Two.
While Bury named an unchanged line-up, Bournemouth axed six players following their FA Cup upset by Blyth Spartans, with two youngsters handed their first senior start and ex-City striker Lee Bradbury playing at right-back.
The changes had the desired effect as midfielder Marvin Bartley burst forward early on and hit a 20-yard strike straight at Bury keeper Wayne Brown.
Richie Baker then conceded a free-kick on the Bury left, from which Matt Tubbs fired wide under pressure from Stephen Dawson.
Bury threatened on the break, as they have done on the road all season, and Dawson blazed over from their first shooting opportunity in the 20th minute.
The hosts replied with Tubbs testing Brown before Carl Preston almost connected with a low cross by Connell, who got in behind Dave Buchanan.
But Bournemouth grabbed a deserved lead in the 39th minute after Elliott Bennett was dispossessed in midfield.
Buchanan cleared Preston's deep cross as far as Igoe, whose low strike found the net despite Efe Sodje's attempted block.
And the Cherries soon doubled their lead as Partington crept into the box unmarked to sidefoot Tubbs' left-wing cross past Brown from 16 yards.
Bury made two changes at the break and substitute Andy Morrell soon called the home keeper into action.
Andy Bishop stung the hands of Shwan Jalal before the home keeper needed to tip over a 25-yard strike by Baker.
Buchanan then did superbly to release sub Dean Howell late on, only for Bishop to scuff Howell's cut-back harmlessly wide.
Tweet
