The Everton faithful briefly thought they had made the perfect start to the new season when Jack Harrison tucked the ball home from close range, only for his celebrations to be cut short by the assistant’s raised flag.
After surviving that early scare, Brighton offered a positive response through Joao Pedro, whose long-range effort rattled the frame of Jordan Pickford’s goal.
The woodwork could not save the Toffees a second time however, as debutant Yankuba Minteh drove the ball across goal for Kaoru Mitoma to tap home a 26th-minute opener for the Seagulls.
That sent the hosts into the break behind, but a reprieve looked to be on the way soon after the restart when Lewis Dunk’s challenge on Dominic Calvert-Lewin saw Everton awarded a penalty.
However, their joy soon turned to frustration as the decision was overturned on review - setting the tone for a second half of misery for the Toffees.
A loose pass from Idrissa Gueye was punished severely, with Danny Welbeck producing a precise left-footed finish to score in a 15th consecutive Premier League campaign and give Brighton a two-goal advantage.
To rub salt in Evertonian wounds, the experienced Ashley Young was given his marching orders for the fourth time in his Premier League career after being penalised for pulling back Mitoma as the last defender.
With a numerical advantage in their favour, the Seagulls added to their tally through Simon Adingra’s 87th-minute effort - the 22-year-old cut inside onto his left foot before slotting the ball into the bottom corner.
That well and truly put the seal on Brighton’s first Premier League away win since February, while inflicting the worst possible start on Sean Dyche’s Everton.
Flashscore Man of the Match: Kaoru Mitoma (Brighton)