Stanley will probably not have a better chance to pick up three points having led for more than 70 minutes.
They took an early lead when David Brown went down under the slightest pressure from Bury defender Chris Brass.
The referee had no hesitation in awarding a penalty and Andrew Todd stepped up to the plate to fire to perfect penalty home.
That was the highlight of a first period in which both sides struggled to maintain a rhythm in a game littered with stoppages.
Home keeper Ian Dunbavin had to go full stretch to tip over an effort from David Buchanan but he was generally well-protected by his defence as the visitors turned the screw after the break.
Accy were left to rue their failure to close the game out when they had the chances in the early stages of the second half.
Skipper Paul Mullin hit tamely at the keeper when he should have done better and then Andrew Proctor completely fluffed a shot from close range with the open goal gaping.
Andy Bishop became more prominent for the Shakers in the second period.
He had a shot charged down before setting up a chance for Glynn Hurst following a strong run.
He then played in Marc Pugh who was denied by a good tackle before Bishop himself finally breached the defence for an equaliser which had been on the cards for some while.
A Buchanan corner was knocked back across the face of goal by substitute Matt Blinkhorn and Bishop was on hand to finish from close range.
The home side had to withstand intense pressure in the closing stages as Bury went in search of the winner. Blinkhorn fired in a low shot through the crowded area but the Accrington keeper went full length to make the save.
Paul Scott and Hurst both flashed headers narrowly over the bar before Bishop had the final chance but his low shot produced another good save.
Eventually the Shakers had to admit that the winner was not going to come and Stanley were able to hold on to what was a deserved point given the determined defensive efforts superbly marshalled by Philip Edwards that had been put up.