Mark McGhee's strugglers are nine points adrift of safety with six games to go after their 17th draw in 40 matches, while their fellow strugglers all won.
It all seemed to be going according to plan for Brighton on a Withdean pitch turned into a mud bath by heavy rain when Noel-Williams nodded them into an 18th minute lead.
The Burnley centre-forward, signed on loan for the rest of the season, steered a header from an Adam Hinshelwood cross beyond Luton keeper Marlon Beresford.
It was Noel-Williams' first goal since August and completed a hat-trick debut, having scored previously on his first appearances for both Burnley and Stoke.
Luton only threatened an equaliser on a couple of occasions in the first half.
Brighton keeper Wayne Henderson parried an angled shot from Paul Underwood when the Hatters midfielder had time to turn and shoot from inside the area.
Two minutes later Ahmet Brkovic headed wide when Warren Feeney nodded back a deep cross from Sol Davis.
Dean Hammond had an opportunity to extend Brighton's lead following good work by Noel-Williams and Colin Kazim-Richards but his shot was blocked away.
That miss proved costly as Luton levelled on the hour when Henderson could only palm away an angled drive from Feeney.
The ball ended up in the back of the net as Steve Robinson and Paul McShane challenged.
Gary Hart came agonisingly close to grabbing Brighton's first win in 11 matches with 17 minutes remaining.
His volley struck the far post via a defender's shoulder after substitute Joe Gatting set him up.
Noel-Williams went close again three minutes later, finding the side netting from an acute angle as Gatting tried to cushion a Richard Carpenter cross into his path.