Carlton Morris scored in the 81st minute to set up a grandstand finish, but Brighton netted twice more in the final exchanges to ensure that they started their campaign with an impressive win.
James Milner, Mahmoud Dahoud and Pedro were all named in the Brighton XI for their debuts, while Luton had six summer signings - Thomas Kaminski, Mads Andersen, Issa Kabore, Tahith Chong, Ryan Giles and Marvelous Nakamba - in their starting side at the Amex.
Tom Lockyer was also back to captain the visitors after collapsing during the Championship playoff final in May, with the defender lining up in a back three.
Lewis Dunk headed wide of the Luton goal inside the opening three minutes, before Morris's dangerous cross was gathered by Seagulls goalkeeper Jason Steele down the other end, with both sides showing their attacking intent in the early exchanges.
Pedro passed up a brilliant chance to score on his debut early on, with the forward firing a low cross from Pascal Gross wide of the post.
Luton had an excellent opportunity of their own in the 23rd minute through Elijah Adebayo, but he could only head wide from inside the box under pressure.
It was Brighton that took the lead in the 35th minute of the contest, though, with March finding the back of the net with a brilliant header after meeting a cross from Mitoma.
Luton had a chance to level shortly after, but Steele did brilliantly to deny Morris, before Danny Welbeck hit the post down the other end during a lively end to the first 45 minutes.
Brighton almost doubled their lead early in the second half when Gross struck one from distance, but his effort hit the outside of the post, with the visitors surviving a dangerous moment.
Luton were impressing defensively, but the promoted outfit were finding it difficult to make their mark down the other end, which could be a big problem this season.
Brighton were awarded a penalty in the 69th minute when Lockyer brought Pedro to the ground, and the Brighton debutant took on the responsibility from the spot, finding the back of the net to double his team's advantage in the 71st minute.
Luton had a penalty of their own in the 79th minute, with Dunk penalised for handball, and Morris stepped forward to score to set up a grandstand finish at the Amex.
Brighton made sure of the points late on, though, with Adingra volleying into the back of the net from close range after a terrible mistake from Nakamba, who lost the ball inside his own box.
Ferguson also struck the inside of the post, before Gross missed a brilliant opportunity late on, but there was time for a fourth deep into stoppage time, with Ferguson converting from close range.
Luton will be bidding to bounce back from the defeat when they travel to Chelsea on August 25, while Brighton will visit Wolverhampton Wanderers next Saturday.