The Stags had John Dempster sent off in the 64th minute but held on to extend the Shakers' winless run to nine games.
The visitors dominated the first half and debutant Jack Blake was denied an early goal by a superb full-stretch save from Brian Jensen, who also denied Colin Daniel when one-on-one with the Mansfield man.
Stags goalkeeper Alan Marriott ensured it remained scoreless at half-time by tipping Shaun Harrad's stinging shot behind.
The second half started in a cagey fashion but exploded into life when Dempster was sent off for a two-footed challenge on Harrad.
Bury instantly made use of their man advantage but Danny Mayor headed Shaun Beeley's cross over the crossbar from point-blank range.
Marriott made a good one-handed save to keep out a 25-yard effort from Tom Soares and Danny Hylton headed another great chance over the bar for the hosts.
Mansfield boss Paul Cox was then sent to the stands in the seventh minute of injury time, ending an eventful game.