The Seagulls dominated possession and registered 25 shots, with six on target, but struggled to break down the hosts’ disciplined low block.
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Ibrahim Osman, Georginio Rutter, Yasin Ayari and Luka Vuskovic all went close, while Tromso goalkeeper Jakob Haugaard produced several impressive saves.
Brighton goalkeeper Bart Verbruggen was largely untested but made a crucial second-half stop from Heine Asen Larsen after the forward beat the offside trap.
Tromso defender Isak Vadebu later missed a golden opportunity to win the match, firing wide with only Verbruggen to beat.
Brighton will now look to finish the job at home and secure their place in the Conference League group stage.