Last year's Conference champions twice went behind, only to battle back and give the Shakers plenty to ponder.
Strike partners Andy Bishop and Glynn Hurst led the way for the home side, both netting first-half goals which put Chris Casper's charges in front.
But equalisers also in the first half from leading scorer Gary Roberts - his seventh of a successful campaign thus far - and then a goal from Romuald Boco deservedly levelled matters to guarantee a share of the spoils.
Casper made one change from the side which drew 2-2 with Barnet with midfielder Daryl Taylor dropping to the bench and Nicky Adams brought in to bolster the Shakers' firepower as the home side went in search of three points.
And it was from the first attack of the game that Bury scored the opener.
Left-back Tom Kennedy whipped in a testing corner and, when Accrington failed to clear the ball in their own six-yard box, Bishop wasn't go to miss from close range.
The former York City striker fired into the roof of the net for his third goal of the season.
But the home side's joy was short-lived when, just five minutes later, Richie Baker carelessly gave the ball away just inside his own halfway line.
Stanley midfielder Ian Craney pounced and, following a bursting run, found Roberts with a neat pass. The former Liverpool schoolboy's powerful left-foot shot flew under the body of the despairing Kasper Schmeichel between the Bury posts.
Hurst saw a left-foot strike parried and saved at the second attempt by Stanley keeper Ian Dunbavin, before Robert Williams had a looping header saved at the other end.
Midfielder Marc Pugh saw a powerful free-kick tipped round the post moments later as Bury got on top, before Hurst had a penalty appeal turned down by Lancashire referee Graham Salisbury after Dunbavin felled him in the box.
However, the lead was restored when Hurst leaped above Dunbavin to head home Kennedy's cross from the left.
But a minute later it was all square as Boco fired under the body of Schmeichel.
There were plenty of chances following the resumption, with Bishop unlucky not to give the home side their third lead of the game.
However, Dunbavin made an excellent point-blank save and, when in the 73rd minute Hurst crossed from the left, the keeper spectacularly saved with his feet to deny Baker's header, before smothering the ball at the feet of substitute David Buchanan who badly scuffed his shot.
Stanley nearly won it through Craney, but his 30-yard thunderbolt was brilliantly saved by Schmeichel who made amends for his earlier blunder.