The Swans, on a 10-match unbeaten run in all competitions heading into the relegation six-pointer, fall back below the dotted line due to their inferior goal difference compared to the two sides directly above them.
Glenn Murray was the chief architect for the Seagulls' crucial home win, scoring twice to put his side two goals in front, before Anthony Knockaert and Jurgen Locadia netted either side of a Lewis Dunk own goal.
Brighton's opener arrived with 18 minutes on the clock, with Murray converting the first of his two goals from the penalty spot following Mike van der Hoorn's clumsy challenge on the striker.
Visiting boss Carlos Carvalhal, who had seen his side draw three away games in a row prior to today, decided to change things around with 10 minutes of the first half to play by bringing on Andre Ayew for Nathan Dyer.
The early change did not have the desired effect, though, as Brighton continued to probe for a second and came within inches of finding it after Duffy climbed highest to send a header against the crossbar.
Swansea were themselves denied by the frame of the goal moments later, however, with Jordan Ayew cutting inside and getting the better of Ryan, only for the upright to keep him out.
Referee Mike Dean then waved away a strong appeal for a City penalty after the ball appeared to hit Murray on the hand inside the box, and the officials made another big - and correct - call seven minutes later to rule out Murray's tap-in for offside.
Brighton were beginning to turn the screw and were rewarded with the game's second goal, as Murray made it seven goals in nine matches by converting from Izquierdo's pull-back.
Knockaert added the third after being played through on goal by Murray, though the Welsh outfit did have a lifeline of sorts as Tammy Abraham's shot found a way past Ryan via a deflection off the unfortunate Dunk.
There was still time for Brighton to restore their three-goal cushion late on, however, thanks to a simple close-range finish for Locadia on his Premier League debut.
Brighton, who had never achieved a league double against Swansea, move up to 11th place and just one point outside the top half.
BRIGHTON & HOVE ALBION (4-4-1-1): Ryan; Bong, Dunk, Duffy, Schelotto; Stephens, Propper, Knockaert (March 79'), Izquierdo; Gross (Kayal 91'); Murray (Locadia 82')
SWANSEA CITY (5-4-1): Fabianski; Naughton, Van der Hoorn (Narsingh 46'), Fernandez, Mawson, Olsson; Dyer (A.Ayew 36'), Clucas, Carroll (Abraham 66'), Ki; J.Ayew