The 26-year-old found the net within the first minute to hand the Hatters a dream start in Bedfordshire, before Chiedozie Ogbene's second league strike of the term doubled the hosts' lead just 120 seconds later.
Luton went into the half-time break three goals ahead following Adebayo's second of the match on the 42nd minute, and shortly after the restart the former Walsall man had sealed his long-awaited hat-trick in first-team football.
Having now earned three wins from their last five Premier League matches, the Hatters finally have their heads above water in the scrap for survival, leapfrogging Sean Dyche's Everton into 17th place.
Following back-to-back goalless draws in Brighton's last two Premier League contests against West Ham United and Wolverhampton Wanderers, supporters may have been expecting a cagey start to a vitally-important clash for both sides, however those of a Hatters persuasion were soon on their feet in celebration.
With the Seagulls playing backwards from the opening whistle, Edwards's side were able to win the ball courtesy of a well-organised press and find Ogbene on the right flank, who sent an inviting cross into Carlton Morris at the back post, with the 28-year-old nodding down to Adebayo who provided the touch to beat Jason Steele.
The home supporters had barely finished settling down following the opener when their Bedfordshire heroes doubled their advantage in the third minute, when a sleepy Brighton backline failed to deal with the run of right-wing-back Ogbene, who had been sent through by a lovely-weighted ball from Albert Sambi Lokonga.
After finding the net in the FA Cup success over Bolton Wanderers earlier in the month, the former Rotherham United man Ogbene was in a clinical mood as the Seagulls' Steele raced towards him to close the angle, with the Irishman tapping the ball past the onrushing goalkeeper before finishing into an empty net.
Finding themselves two goals behind, a shell-shocked Brighton posed greater threat as the half developed, largely through Brazilian starlet Joao Pedro - whose Watford connections did not go unnoticed by the Hatters faithful - although Roberto De Zerbi's men were unable to sting the palms of Thomas Kaminski.
Just three minutes before the break Brighton's woes worsened, with the Seagulls being dispossessed in their own half once again by Barkley, who found the run of Adebayo on the right side of the goal, and the Hatters' leading marksman did the rest, beating Steele at his near post with a calculated finish.
Seemingly furious with his side's first-half showing, De Zerbi made two alterations for the second period, with Evan Ferguson and Tariq Lamptey replacing Pervis Estupinan and Facundo Buonanotte, who had been tormented down his flank by the marauding Ogbene on countless occasions in the opening 45 minutes.
The Seagulls' changes did not shift the momentum of the contest, though, and Luton's number 11 was able to secure his maiden hat-trick in professional football, with the striker heading in from close range following an inviting Alfie Doughty corner, which was helped into the path of the goalscorer by Pedro's flick-on.
Brighton have now failed to score in four of their last six Premier League matches, with the South Coast club currently unable to translate their potent habits from the FA Cup, a competition in which they have netted nine goals in two games away at Stoke City and Sheffield United.
Buoyed by their biggest top-flight success since the 1980s, Luton will travel to Eddie Howe's Newcastle United on Saturday afternoon, when the Hatters will have the chance to extend their winning run across all tournaments to four matches following an unbeaten January.
The Seagulls are also back in action on Saturday as they host rivals Crystal Palace, with De Zerbi's side yet to collect a Premier League maximum in 2024.