Brighton came storming back and dominated for much of the second half but struggled to convert their possession and slick passing into goal-scoring chances until Maupay threw himself into the air to fire home Tariq Lamptey's cross from the right in the 89th minute.
Despite having to reshuffle their pack, Brighton came storming back and Jakob Moder should have put them level after he was teed up by Maupay just before the break, but he fired his effort into the legs of Lukasz Fabianski.
The visitors opened the second half with two more good chances for March and Lewis Dunk before West Ham had another goal from a corner ruled out for offside after a lengthy VAR review.
With West Ham looking to hit on the counter-attack, Brighton keeper Robert Sanchez pulled off a brace of superb saves to keep his side in it, with one full-length diving effort to deny Declan Rice bringing roars of approval from the travelling fans.
Brighton continued to dominate possession but struggled to convert their slick passing into goal-scoring chances until Maupay threw himself into the air to fire home Tariq Lamptey's cross from the right in the 89th minute to snatch a point.
The draw stretches Brighton's streak to nine league games without a win and leaves them seventh in the table on 19 points, while West Ham remain fourth on 24 points.