This was a battle at the bottom of the table and the Shakers should have ran out comfortable winners after Lawrie Wilson was sent off in the 31st minute.
However, they could not make a chance count and Accrington will have been happier with the point.
Stanley had the first real opening when winger Lee Molyneux forced Bury keeper Brian Jensen into an early save.
Bury had more of the early possession and Anton Forrester should have opened the scoring after 15 minutes when he broke through but the legs of Stanley goalie Marcus Bettinelli denied him.
Stanley were down to 10 men when Wilson dived into a challenge on Bury captain Craig Jones and referee Kevin Wright had no hesitation in sending the left-back off.
It was then wave after wave of Bury attacks until the end of the half with Forrester guiding a free header wide while Nicky Hunt cleared off the line from Jones for the home side.
Stanley re-arranged their team at the break and had more of an impact after the re-start with Molyneux's fierce goalbound strike hitting Pablo Mills while Bury's Forrester got in a good position a number of times but could not find the finish.
Bettinelli produced a superb save from a tight angle to deny Jones at the death and also kept out a Daniel Nardiello goalbound strike as the 10 men held on for a point.