Tromso goalkeeper Jakob Haugaard produced two sharp first-half saves to deny Georginio Rutter and Ibrahim Osman as Brighton controlled much of the contest but failed to find a breakthrough.
The hosts threatened on the counter-attack as Heine Larsen was foiled by Bart Verbruggen, before Isak Vadebu wasted a golden opportunity in the closing minutes for Tromso.
Seagulls manager Fabian Hurzeler handed defender Luka Vuskovic his debut following his club-record move from Tottenham, but was without Carlos Baleba and Yankuba Minteh. Both players are sidelined through injury but have been linked with moves to Manchester United and Liverpool respectively.
Brighton are appearing in Europe for only the second time after reaching the Europa League last 16 in 2024. They will host Tromso next Thursday in the return leg.
They first open their Premier League campaign at home to Aston Villa on Sunday.