Quinn scored with an angled drive in the 84th minute to extend the Cobblers' undefeated sequence on the road to a club record 13 matches stretching back to March.
The Northern Ireland international made the most of a quickly taken throw-in by the visitors after Brighton defender Adam Hinshelwood had cleared over the stands.
The makeshift home side looked like hanging on for an unlikely victory until Quinn's strike.
Northampton started the match well and should have taken the lead in the 13th minute.
A defence-splitting pass by David Hunt released Mitchell Cole and he rounded goalkeeper Wayne Henderson before squaring across the box, only for skipper Scott McGleish to miss a gaping goal from ten yards.
Brighton improved as the first half wore on and they went ahead in the 39th minute.
A corner was cleared as far as Dean Cox, lurking just outside the box. His well-struck volley was blocked close to the line by team-mate Jake Robinson, who then turned sharply to find the net from point-blank range.
Robinson threatened to make the game safe for Brighton in the second half. He had a shot which swerved narrowly wide and was then denied by Chris Doig's tackle after latching onto a pass from Tommy Fraser.
Brighton strikers Alex Revell and Sam Williams were ruled out with respective back and groin injuries and the Seagulls' injury crisis deepened in the second half.
They lost skipper Dean Hammond with a groin injury, while on-loan Coventry defender Andrew Whing was taken to hospital for stitches in a bad eye wound following a collision with Northampton goalkeeper Mark Bunn in the closing stages.
Brighton had to play out five minutes of stoppage-time with only ten men, having already used all three substitutes.
Northampton almost snatched victory from a smartly taken free-kick in time added on. Quinn had a shot smothered by Henderson and Andy Kirk's follow-up was deflected just wide.