The Seagulls had the better of the chances throughout, with Tromso goalkeeper Jakob Haugaard called into action twice in the first half to deny a low strike from Georginio Rutter and a volley from Ibrahim Osman. Despite Brighton's dominance in possession, the home side had two clear openings on the counter-attack.
The first arrived in the second half when Heine Larsen beat the offside trap and found himself one-on-one with Bart Verbruggen, only for the Brighton goalkeeper to block with his body. In the 85th minute, Isak Vadebu also got in behind Brighton's high defensive line but skewed his effort wide when clean through on goal.
Manager Fabian Hurzeler said his side controlled the tie without doing enough to break down a well-organised opponent.
"I think we controlled the game but we weren't quick enough in possession," Hurzeler said. "We still had a few opportunities, but I think overall Tromso defended well.
"They had two big chances after transition moments where we have to defend better. Overall a tough opponent, which is what we expected.
"I think we expected exactly how Tromso played. They defended well, very compact. They had a lot of players in their box who blocked the shots well, and they had a really good goalkeeper and were good in transition. We created opportunities, but in the end we have to be satisfied with a draw, although it's not the result we hoped for."
Brighton, who are playing in a European competition for only the second time having reached the Europa League last 16 in 2024, will host Tromso in the return leg at the Amex next Thursday. Hurzeler expects a tough test even with home advantage.
"Now we are looking forward to the opportunity to win the game in front of our own fans," he said. "There are certain aspects we have to improve against Tromso. Overall, like I said, they are a good football team so we will have to play on our highest level to win the game at the Amex."
Brighton dominated the ball throughout the trip to the Romssa Arena but were unable to find a breakthrough against a Tromso side who defended in numbers and made the most of their rare forward opportunities.
The result leaves the tie finely poised heading into next week's second leg, with Brighton needing a win or a scoring draw to progress to the group stages of European football's third-tier competition.