Corr and Will Atkinson both struck in the first half as the Shrimpers eased to a first win in three games that leaves them eight points off an automatic promotion place having played a game less than the top three.
Adam Thompson was replaced by Cian Bolger after just 10 minutes due to a shoulder injury, but it did not hold back the hosts.
Corr fired them into the lead in the 21st minute when he was picked out by Michael Timlin before lifting the ball over Lenny Pidgeley.
The lead was doubled six minutes later when Atkinson bagged his first goal of the season after Pidgeley parried Ryan Leonard’s shot into his path.
Mansfield had no answer in the second half as they slipped to a third consecutive defeat that sees them sitting just six points above the relegation zone.
The closest they came was through Vadaine Oliver’s header which came back off the crossbar, while substitute Rakish Bingham was denied by Daniel Bentley.