Maupay’s overhead kick a minute from time means West Ham are still waiting for a first Premier League win over the Seagulls after nine attempts.
The Hammers looked to have finally beaten their bogey team as they led through Tomas Soucek’s first-half header.
But Brighton had other ideas and finally put one of their chances away to deny the hosts victory yet again.
This had the feel of a big match in the context of West Ham’s lofty aspirations this season, having clung on to fourth place despite successive defeats on the road against Wolves and Manchester City – and with leaders Chelsea the visitors on Saturday.
Alarm bells may have started to ring had Brighton taken the lead inside two minutes, but Maupay fired across goal and wide from close range after a defensive mix-up allowed Yves Bissouma to stroll into the penalty area.
Instead, West Ham went ahead in the fifth minute through a familiar routine.
Seagulls boss Graham Potter admitted beforehand that he was wary of West Ham’s threat from set-pieces, yet from the first corner of the match Pablo Fornals crossed to the near post where Soucek was allowed to leap unchallenged and glance the ball into the net.
It was almost two when Michail Antonio headed Vladimir Coufal’s cross back across goal to Fornals, whose volley crashed against the underside of the crossbar.
Brighton’s cause was not helped by losing two players, Jeremy Sarmiento and Adam Webster, to injury in the first half.
But one of their substitutes, Solly March, almost created an equaliser when he played in Maupay, who in turn fed Jakub Moder in front of goal, but his shot was well saved by fellow Pole Lukasz Fabianski.
The problem in the first half was as stark as it had been in the 0-0 draw against Leeds on Saturday, when they were booed by some of their own fans – Brighton simply could not find the net.
After the break March had a cross deflected over and Lewis Dunk headed the corner straight at Fabianski, before West Ham were denied a second by a lengthy VAR check.
It looked like a job for cricket’s snickometer to prove for certain whether the ball had brushed the ankle of the offside Antonio as it was bundled in from a corner, but the goal was eventually chalked off.
Jarrod Bowen squandered a chance to kill Brighton off when his shot rolled past the far post and late Robert Sanchez saves kept Declan Rice and Antonio at bay.
They were to prove costly as the clock ticked down and Brighton sub Tariq Lamptey stood up a cross which Maupay expertly hooked home to frustrate the Hammers yet again.