Solly March and Joao Pedro had put the Seagulls 2-0 ahead with 20 minutes to go, as although Carlton Morris pulled one back from the spot, substitutes Simon Adingra and Evan Ferguson netted late on to seal the victory.
To see the best pictures from the Amex Stadium, check out the gallery below.
1. Brighton & Hove Albion 4 Luton Town 1
Ryan Giles, John McAtee and Luke Berry inspect their new surroundings at the Amex Stadium ahead of kick-off. Photo: Charlie Crowhurst
2. Brighton & Hove Albion 4 Luton Town 1
A Luton fan with his flag ahead of kick-off at the Amex Stadium. Photo: JUSTIN TALLIS
3. Brighton & Hove Albion 4 Luton Town 1
Two Hatters fans await Luton's first ever Premier League game on the south coast Photo: JUSTIN TALLIS
4. Brighton & Hove Albion 4 Luton Town 1
Seagulls boss Roberto De Zerbi greets Hatters boss Rob Edwards ahead of kick-off. Photo: JUSTIN TALLIS