Kauno were entering new territory, with their first-ever European main draw match following their maiden league title last term and coming off the back of their first Lithuanian Football Cup triumph four days ago.
There was no time for their heads to be in the clouds, though, as Brighton dominated possession and relentlessly got crosses into the box with Costinha heading over.
Three blocks in quick succession were required after a corner, and Tomas Svedkauskas tipped over a Pascal Gross free-kick before Michael Svoboda looped a header onto the roof of the net.
The visitors tried to make forays forward of their own, and Franco Baldassara sent a strike wide from the edge of the box, but Brighton were soon back on the ball as their quest for an opener continued.
Lewis Dunk resorted to a curling shot from range on the half-hour mark that grazed the roof of the net on its way over, before Jack Hinshelwood saw his bouncing header tipped away by Svedkauskas.
Brighton eventually broke the deadlock in the 41st minute, as Ibrahim Osman kept the ball in by the finest of margins and cut it back for Georginio Rutter to pick out the bottom corner.
Seconds before the half-time whistle, Rutter turned provider by fizzing the ball across for Hinshelwood to convert on his return from injury.
Both managers made a double substitution ahead of the restart, and Luka Vuskovic headed off-target shortly after his introduction.
Svedkauskas then denied another Hinshelwood header, but he was beaten in the 53rd minute when a swift Brighton move culminated in Osman squaring for Rutter to score into the vacant net.
The visitors suffered yet more blows while Brighton had reason to celebrate, with Amine Benchaib going off injured before 18-year-old Younes Ibrahim came off the bench and immediately delivered a beautiful cross that Matt O’Riley headed home his first Seagulls goal since August 2025.
That was the end of the scoring despite Costinha getting more headers off, as Brighton comfortably earned their fifth consecutive win across all competitions for the perfect start to the Conference League campaign in which they are deemed to be the favourites.
In doing so, they inflicted just the second defeat on Kauno in their last 10 matches across all competitions.
Flashscore Man of the Match: Georginio Rutter (Brighton & Hove Albion)