The former Liverpool defender succeeds Oscar Garcia at the Championship club after the Spaniard quit following their play-off defeat to Derby in May.
Hyypia, 40, was given his first job in management by Bayer Leverkusen in May 2012, but the Finn was sacked last April after a poor run of form threatened their qualification for the Champions League.
"I am absolutely thrilled by this appointment; Brighton & Hove Albion are an ambitious club and I am very pleased to have been given this opportunity," Hyypia told the club's website.
"I've held extensive talks with the chairman Tony Bloom and chief executive Paul Barber. I have looked at the club from top to bottom and I am very impressed by what I've seen at all levels.
"I've seen the Amex Stadium and the club's new training ground and they are amongst the best I've seen anywhere in the world.
"I've also already met some of the key staff including Nathan Jones, the club's head of football David Burke, and Dr Helge Riepenhoff, head of medical and sports science, and I am very much looking forward to working with all of them."
Chairman Tony Bloom revealed Hyypia was the stand-out name from what he called a "wealth of very high-calibre applicants" and has backed the new manager to succeed, with promotion to the Barclays Premier League likely to be the remit.
"We are very excited to have appointed Sami as our new manager," said Bloom.
"He has already had a distinguished career in the game, having played and managed at the very top. As we have said previously we had a wealth of very high calibre applicants.
"Paul and I undertook an extensive and diligent evaluation of the contenders and Sami was the one who emerged as our No.1 choice.
"I would like to thank the fans for their patience during this time. I know the fans were keen to see a new manager in place quickly, but it was vital to take our time.
"I am extremely confident that Sami will be a big success at Brighton, and I am sure the fans will share our delight at his appointment."