Shrimpers' defender John Egan's late header was the only effort either side managed on target all afternoon.
And out of form Southend were forced to play the final eight minutes of the match with 10 men after Barry Corr was sent off for a tangle with Shakers defender Frederic Veseli.
Southend striker Freddy Eastwood wasted the best chance of the game at the start of the second half when he fired over the crossbar from eight yards in the 56th minute of the match.
Eastwood, who was making his first start since September, immediately made way for Corr and he soon saw red after an off the ball clash with Bury defender Veseli.
Bury changed goalkeepers at half-time with youngster Reice Charles-Cook replacing Brian Jensen.
But none of the shot-stoppers were called in to action too often, meaning Southend have still not won since mid-January.