Newcastle's misery was complete when Neal Maupay's sharp finish made it 3-0 with around 20 minutes remaining.
Sixteenth-placed Brighton's second successive victory put them on 32 points from 29 games, six points above third-from-bottom Fulham who have also played one game more.
Newcastle, without a league win in six games, are 17th and only two points ahead of Fulham, albeit with a game in hand.
It was the sort of night that has summed up Newcastle's season as shortly before going behind, they lost midfielder Isaac Hayden to what looked a serious leg injury after an accidental collision with Brighton's Yves Bissouma.
And just a minute before Welbeck struck they were denied an equaliser when Ryan Fraser's curling effort struck the post.
Newcastle could have few complaints though as Brighton goalkeeper Robert Sanchez did not have to make a save.