The 18-year-old midfielder has penned a development contract and will join the Championship side's youth team.
Asmundssen attracted Brighton's interest while playing for Icelandic team Fylkir, for whom he played six times, scoring once.
He told the club's official website: "It means a lot to me to be joining Albion, as it is a big chance at a big club. The atmosphere here is great and the club is really progressing.
"Compared to the football in Iceland it is a better level, so this is a big step for me and I'm looking forward to the challenge."
Asmundssen is the second young player to have arrived at the Amex Stadium this month, after 22-year-old Latvian international Vitalijs Maksimenko joined from Skonto Riga.
Brighton's academy manager, John Morling, said: "Emil came via the recruitment department and impressed on trial.
"He was playing in the Icelandic league, which is a good level for someone his age, plus he is representing his country at under-19.
"He is a midfielder who has lots of energy and we feel he is a player with a real chance of playing in the first team in two or three years."