It looked like being a frustrating afternoon for Brighton who dominated the game but could find no reply to Wilfried Zaha's 19th-minute penalty until 21-year-old Mac Allister thumped one home in the 90th minute.
Brighton would have gone home scratching their heads had they not levelled after peppering Palace's goal with 20 attempts compared to the one of the hosts.
They have four points from their opening five games while Palace have seven and are in 12th spot.
"I am pleased with the performance and the team," Brighton manager Graham Potter said. "If we play that game 10 times we win it quite a few times."
There was not a great deal of contact but the VAR check agreed with referee Stuart Attwell, and Zaha made no mistake from the spot, firing past Mat Ryan for his fourth goal of the season.
"It looked a little soft. In defence of the referee, as soon as an arm goes up you give them an excuse to give the penalty," Potter said.
Palace showed little ambition after that as they protected their advantage.
He also had a clear sight at goal after halftime but delayed his shot and was closed down.
Brighton kept probing, though, and when Palace defender Joel Ward made a mess of dealing with a long punt in the last minute of normal time, Maupay shifted the ball across to Mac Allister, whose fierce shot clipped a Palace defender on its way into the bottom corner of the net.