The Swans failed to overcome the loss of five first team regulars and fell to a disappointing defeat.
When Kevin Nugent opened the scoring with only 90 seconds on the clock things looked good, but straight from the restart David Nugent ran through and leveled matters.
An unmarked Dwayne Mattis volleyed in to put the Shakers ahead and the writing was on the wall for the makeshift Swans.
Any hopes of rescuing a point was dashed when Brian Murphy struck an 83rd minute free-kick around the wall leaving a static Willy Gueret helpless.
The golden booted Lee Trundle set up Swansea's goal when he turned two defenders and crossed to Nugent who fired home.
Swansea went to sleep straight from the restart and once Bury's Nugent had beaten Alan Tate down the left flank he had a clear run on goal before beating Gueret with a well-placed shot into the corner of the net.
The home side had barely regrouped when Mattis put them ahead and from then on they were always fighting a losing battle.
Newcomers Ijah Anderson and Marc Goodfellow looked well short of match fitness, and with Tate unsure at the back, the Swans always looked likely to concede another.
Bury were well on top in midfield where David Flitcroft was magnificent and, with Danny Swailes keeping a tight rein on Trundle, the Swans rarely looked like clawing their way back into the game.
Murphy put the seal on a good win for the Lancastrians in the closing stages and. despite one or two moments, the Shakers well deserved the third away win of the season.