Hammers had opened the scoring in the sixth minute, when defender Glen Johnson rose to head in a cross from Terry Khan, and then had a number of chances to extend their lead before half-time.
However, the hosts equalised just a minute after the break and seemed to be holding out for a valuable point until the 86th minute, when a goalmouth scramble saw the ball drop to Britton, who fired low into the net from the edge of the area.
Said Tony Carr afterwards: "It was pleasing to win like that after a tough game.
"To be honest, we controlled the game in the first half and should have scored two or three before half-time.
"They came back strong in the second half, though, and had a few big lads who caused us problems at set pieces.
"It was nice to see Leon hit the winner, because he hasn't scored many goals at youth level and it was a timely strike."
Team: Jackson, Khan, Lumsden, Ferdinand, Ward, Johnson, Britton, Saunders (Ritchie 55), Allen, Eastwood (Morris 80), Cole (Fletcher 85). Subs unused: McClenahan, Tattum.