Forward Ayew coolly converted Christian Benteke's through ball 20 minutes from time to push Palace a step close to Premier League survival.
Albion remain the only club in the top four divisions without a victory in 2020 and in deep relegation danger after failing to capitalise on a largely dominant derby display.
The Seagulls' winless league run now stands at eight games following a first top-flight defeat on home soil to their fiercest foe.
Graham Potter's men have now tasted victory just once in their past 14 matches in all competitions and sit on 28 points from as many games ahead of a tough run of fixtures, with Palace eight better off.
For all of the bluster on the field and inevitable taunting in the stands, there was very little to separate the sides during a tight opening period.
Brighton controlled possession and wasted the clearest first-half opportunity through March.
The Seagulls winger was left unmarked inside the Eagles' penalty area 10 minutes before the break after being picked out by Leandro Trossard's centre but inexplicably fired into the side-netting.
Palace, unchanged from last weekend's win over Newcastle, had enjoyed the better of the early chances.
Benteke, buoyed by his first club goal of the season earlier this month, volleyed well wide after a fine through ball from James McArthur, while Cheikhou Kouyate was later smothered by Albion goalkeeper Mathew Ryan.
While warming up close to a corner flag, Schelotto appeared to kick the ball at his opponent after it went out of play, prompting a petty confrontation, a yellow card for the Argentinian and boos for the remainder of the afternoon for Palace forward Zaha.
Roy Hodgson's visitors were twice fortunate not to fall behind in the opening 10 minutes of the second half.
A goal-bound header from Albion captain Dunk following a corner struck team-mate Maupay, before the latter rifled a powerful drive straight at Palace keeper Vicente Guaita.
Albion threw on veteran striker Glenn Murray against his former club in the hope of deservedly breaking the deadlock.
But they were made to pay for their profligacy 20 minutes from time.
Ghana international Ayew was sent racing into the box by a perfectly weighted pass from Benteke and he calmly found the bottom right corner via a touch from Ryan to spark wild celebrations among the sold-out away end, including the detonation of a smoke bomb.
Potter slumped in his seat in the dugout after a goal which sucked belief from the home crowd.
Albion mustered little in the time remaining and would have lost by more had Zaha not struck the left post at the end of a stoppage-time counter-attack.
The late miss counted for little in the end as the Eagles hung on to gleefully celebrate a first away win since November 30.