The Iron created the better chances, with the on-loan forward Nicky Ajose particularly wasteful in front of goal.
Mark Duffy created United's first chance after 13 minutes when he charged down the right wing and played in Chris Dagnall, who rounded Brown but could not pick out a teammate.
Wimbledon's first real opening came after 29 minutes when Christian Jolley slipped the ball to fellow striker Rashid Yussuf in the box, but Sam Slocombe got down low to his left to make a comfortable save.
After the break Brown was forced into a great low save to his left from Ajose's venomous shot after 57 minutes.
Wimbledon's Max Porter turned and hit a snap shot in the area after the ball had gone loose, but his effort rolled past the right post.
Duffy worked a great chance for himself on 75 minutes but his powerful shot was well saved by Brown as the Iron failed to break through.