Both sides rarely threatened throughout, but the Shakers held on resolutely following Murphy’s sending off for handling outside the penalty area.
Lee Clark made five changes to his side following last weekend’s defeat at the hands of Bristol Rovers as he sought to bring about the Shakers’ first league victory at the Crown Oil Arena since October 2007.
Tsun Dai, Nathan Cameron and Phil Edwards came into the team, as well as loan duo Jordan Williams and Rohan Ince, with Callum Styles, Alex Bruce, Craig Jones, Tom Aldred and Callum Reilly all making way.
After a quiet opening to the game, Bury had their first shot of the game on 20 minutes when Nicky Ajose flashed a shot wide of goal from the corner of the box.
But both sides struggled to make their mark on the match, and, as a scrappy first half drew to a close, Ince forced the first save of the game, stinging the hands of Brendan Moore with his low 20-yard drive.
The hosts could have gone ahead immediately after the interval when a sloppy back-pass from Ajose was intercepted by Matt Done, who burst through on goal but was thwarted by a combination of Alex Whitmore and Murphy.
The Shakers’ afternoon took a turn for the worse on 54 minutes, however, when Murphy came to collect a long ball and handled outside of the area.
The referee showed a red card to the Bury shot stopper, who was subsequently replaced between the sticks by debutant Leo Fasan.
With the benefit of a one man advantage, the hosts pushed for a goal but the Shakers’ defence, marshalled by the returning Cameron, stood strong.
Rochdale’s Jordan Williams broke through the Bury defence on 71 minutes and looked certain to score, but was denied by a great save from Fasan, before Ian Henderson skied the rebound.
Despite a late flurry from the home side, the Shakers continued to defend resolutely to hold on for an important point.
Bury will be in action again next Saturday when they welcome Scunthorpe United to Gigg Lane.
Attendance: 4,601 (1,290 visitors)
Bury starting line-up: Murphy, Williams, Leigh, Cameron, Whitmore, Edwards (Cooney 86), Ince, Dai (Reilly 52), Bunn, Ajose (Fasan 54), Beckford
Unused subs: Aldred, Skarz, Ismail, Lowe
Rochdale starting line-up: Moore, Rafferty, Ntlhe (Brown 20), McNulty, McGahey, Cannon, Allen, Williams (Gillam 80), Henderson, Done, Davies (Inman 45)
Unused subs: Collis, Canavan, Keane, Morley