Skipper Nicky Forster was the hero on his comeback from injury.
The 35-year-old striker scored his 16th goal of the season with 17 minutes left to calm the Seagulls' nerves.
They looked doomed when they lost at home last month to fellow strugglers Swindon but a run of four wins and a draw since then has carried them to safety.
Manager Russell Slade, appointed in March, was engulfed by jubilant fans at the final whistle.
The former Yeovil Town chief, who was given the job until the end of the season, now seems certain to keep the post on a permanent basis.
It was a predictably anxious afternoon on the south coast and Stockport, the latest club to go into administration, put up a stern fight.
They had the first chance in the 28th minute when a snap-shot by striker Oli Johnson was shovelled away by goalkeeper Michels Kuipers.
Brighton almost went ahead in first-half stoppage time. Substitute Calvin Andrew dived to head against the bar after Dean Cox had an effort blocked by Stockport keeper Conrad Logan.
Andrew, on for the injured Gary Hart, hurt himself in the process, which led to Forster's second-half introduction.
It was Forster's first appearance for six weeks following a knee injury which was initially expected to rule him out for seven months.
He almost scored in the 57th minute when he twisted to loop a header just wide from a cross by Andrew Whing.
Forster was then in the right place at the right time when a shot from on-loan Stockport midfielder Gary Dicker, given permission to play for Brighton by his parent club, was spilled by Logan.
Forster knocked in the loose ball from point-blank range to keep Brighton up and send Northampton down.