Bury found Accrington goalkeeper Marcus Bettinelli in inspired form, as the League Two strugglers failed to score.
The home side played an hour of the match with 10 men, after Lawrie Wilson was dismissed for a foul on Shakers captain Craig Jones.
Bury's on-loan striker Anton Forrester was denied by Bettinelli, before guiding a free header wide of goal.
Bettinelli produced a superb save to deny Jones and also kept out a Daniel Nardiello goalbound strike.
Bury move four points clear of the relegation zone in 21st place, while their opponents lie one spot ahead.
VIEW FROM THE DRESSING ROOM
Bury manager David Flitcroft told BBC Radio Manchester:
"We got a point and a clean sheet, but I'm going out of here very frustrated because the sending-off killed the tempo and the momentum of the game. It didn't suit us.
"We tried everything to score and got a lot of forward players on at the end.
"To be fair, you've got to give Accrington credit for the way they defended. They were resolute and determined."
Accrington Stanley 26 Bettinelli 16 Hunt 19 Wilson Dismissed 02 Murphy 06 Winnard Booked 05 Aldred 15 Mingoia 04 Joyce 24 Odejayi (Liddle - 45' ) 09 Webber (Gray - 80' ) 13 Molyneux (Atkinson - 89' ) Substitutes 03 Liddle 07 McCartan 10 Hatfield 17 Gray 20 Windass 25 Atkinson 44 Bowerman Bury 21 Jensen 07 Jones 15 Mayor (Platt - 61' ) 03 McNulty 34 Mills Booked 29 Veseli (Obadeyi - 87' ) 08 Miller 16 Tutte 18 Nardiello 17 Akpa-Akpro (Sedgwick - 45' ) 09 Forrester Substitutes 04 Hinds 06 Proctor 12 Sedgwick 19 Soares 22 Platt 30 Charles-Cook 35 Obadeyi
Ref: Kevin Wright