Bury raced into a 10-0 lead thanks to touchdowns from Brett Marriott and Joe Swain, one of which was converted by Swann.
However, it was Bowling who went into the break 12-10 up following tries from Todd Carter and Jason Marshall together with two Ben Heald extras.
Queensbury’s props Andy Senior and Lee Kelley used their bulk and experience to reach the chalk and Swann slotted over another extra two before Bowling’s second-row forward Richard Lumb posted a brace, both converted by the accurate Heald, to put the game on a knife edge.
Full back Ben Naylor raced 40 metres to level the score, leaving Swann to settle matters in the dying embers of an absorbing encounter which ensures Bury Premier Division rugby next term.
It was a double celebration for the Queensbury club as the second team reached the final of the Pennine Supplementary Cup following an 18-14 semi-final win against visitors Hanging Heaton.
Bury came flying out of the blocks playing up the hill and were rewarded when a cut-out pass put winger Andy Walker in at the corner. Bury also scored first in the second stanza when Dave Thomas came onto a short ball to coast in from 40 metres.
Hanging Heaton staged a recovery before Bury’s Darren Brown took a pass at pace to sprint down the touchline and cut inside the full back to score a belter under the sticks.
Then in a tense finish a floated pass from Leighton Ruff put Danny Sykes in at the corner to seal the game.
Birky ran in tries from Damian Smith, Alan Binns, Karl Robinson (2) and Tom Duff. Matt Honysett completed the tally with six conversions and a touchdown.
Wyke won 44-8 thanks to a fine performance from former Wests forward Ian Wormald, who crossed twice and had an hand in four others.
Lewis Wigglesworth and Martin Tordoff replied for the hosts.