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Ciaran McGuckin struck a stunning opener for the visitors but the Shakers hit back immediately when Kai Evans’ corner went straight in.
Anthony Johnson’s side finished the game strongly but couldn’t quite find a winner, with Evans going close on a couple of occasions.
Ahead of kick-off, Bury announced the return of forward Dom Telford on a permanent move from Gateshead.
Telford scored 14 goals during the 2018/19 season as the Shakers finished second in League Two under Ryan Lowe.
Since then, he has had spells at Plymouth Argyle, Newport County and Barrow.
Nearly five thousand supporters were in attendance at Gigg Lane, plus the BBC cameras, as Bury aimed to pull off an upset.
But the first chance fell to the visitors when Will McGowan sent Ellis Taylor down the left wing and he dragged an effort wide.
Bury’s first sight at goal came from Mo Touray, who curled a shot over from the edge of the penalty area.
McGuckin went close after a dangerous cross from Josh Hmami, but the forward couldn’t keep his volley down from close range.
Hmami curled an effort wide as the visitors started to apply pressure, and the opener came from a moment of magic five minutes before the break.
McGuckin cut inside from the left flank and curled a superb finish over Mitch Allen into the top corner from the edge of the box.
But Johnson’s men weren’t behind for long.
Evans’ in-swinging corner went straight in after causing a mix-up between the defence and Bury-born goalkeeper Luke Hutchinson.
Buoyed by the equaliser, the Shakers started the second half brightly and Bailey Sloane had a header saved after getting on the end of Owen Robinson’s cross.
Taylor whistled a strike just over the bar and Shawn McCoulsky was denied by Allen as the visitors looked to regain the lead.
Evans’ curling effort looked destined for the far corner but Hutchinson, on loan from Bolton Wanderers, made a fine save to his left.
The wide man continued to cause problems and had the goalkeeper beaten with a powerful strike from distance, but it flew just wide.
McGowan had a couple of chances at the other end, initially firing over the bar and then forcing Allen into a smart stop to his right.
Evans had one more effort in the final stages, bending a shot just wide as Hutchinson scrambled to his left.
There was ultimately nothing to separate the sides and they will meet again on the other side of the Pennines on Tuesday night.