The Reds kept our side of the final-day bargain by collecting three points on the south coast but Newcastle United's 4-2 triumph at Brentford kept the Magpies ahead of us on goal difference.
It means we have to beat Manchester City in next Saturday's Emirates FA Cup final to obtain European football next term, in the Europa League.
Diogo Dalot and substitute Rasmus Hojlund grabbed the goals in a much-improved display after the break to ensure Erik ten Hag's men approach the big day at Wembley on the back of successive victories.
FIRST HALF - FLAT IN THE FINALE
A largely listless opening 45 minutes saw Brighton carve out the better chances but there was little to enthuse the Amex Stadium supporters.