Leandro Trossard’s curled finish into the bottom corner in first-half stoppage-time was the least Brighton’s early dominance deserved, and Danny Welbeck’s sublime strike from outside the box doubled the advantage six minutes after the restart.
Neal Maupay then put the result beyond any doubt as he met Pascal Gross’ cross for his first goal in 10 league games midway through the second half.
Newcastle were second best throughout and rarely threatened a comeback – they had only one shot on target – though Ryan Fraser was inches away from an equaliser when he fired against the post shortly after the break.
While 18th-placed Fulham missed the chance to move above Steve Bruce’s side with a 2-1 defeat by Leeds on Friday, Newcastle remain in serious trouble with just two points separating them from the relegation places – albeit with a game in hand on the Cottagers.
Brighton manager Potter said victory over Southampton last weekend had given his side “renewed belief” of retaining their Premier League status.
The manner of this comprehensive win against a club directly below them in the table should convince his side that they are more than capable of securing a fifth consecutive season in the top flight.
Battling for survival has become familiar territory for the Seagulls, whose largest final margin to the bottom three since their 2017 promotion is seven points, and there was little sign of pressure in their play as they took command of a crucial game.
An inability to capitalise in front of goal and convert impressive performances into points has kept Brighton from mid-table security – a trait evident in the opening exchanges when 21-year-old Poland midfielder Jakub Moder failed to convert two early chances in his first Premier League start.
However, two expert finishes from Trossard and Welbeck either side of the interval left the visitors dejected, before Maupay added the gloss from close range to secure a second 3-0 win over Newcastle this season.
Having impressed against Saints with arguably his best performance since joining on a free transfer last summer, Adam Lallana also shone in a central midfield role with incisive passing through the Newcastle lines.
Brighton’s fate is now firmly in their own hands, but tough games against Manchester United, Everton and Chelsea await after the international break.