Harwich were dangerous in the opening stages with Harry Tuck making progress down their right flank but Brantham came back with an even first half looking likely to remain goalless.
However, in the last minute of added time, Brantham pressed and took the lead with a shot from close range by Liam Battersby.
With 70 minutes gone, Harwich raised their game, but with games in hand Brantham have the chance of a high finish in the league, and they proved tough opposition.
Shrimpers boss Michael Clarke was making good use of his substitutes with Dylan Crowley (pictured) coming on and pepping up their attack.
When a pass from Liam White found Crowley, he equalised with a really well-taken shot from an acute angle into the far corner of the net.
Ewan Livingston was working hard for Harwich, who were now playing their best football as they scented victory.
The Shrimpers were pushing men forward and Athletic were seeking to catch them out with quick breaks.
So it was that with ten minutes remaining, Harwich were stretched and forced into desperate defending near their goal line that resulted in them conceding a penalty that was well struck by Joe Woods for the winning goal.
Harwich team: Thomas Maurer, Alex Pryke, Macit Batuham, Archie Corner, Joe Simmons, Jamie Beecham, Harry Tuck, Freddy Bethell, Liam White, Cody Harrison and Ewan Livingston. Substitutes: Dylan Crowley, Solomon Griffiths, Rory Breen, Jack Milne and Oscar Warnes.